1 1 


\ 

■) 

; 

/■  > 


■') 


THEIUmiLSElSOFSTilllOllinOL-B 

COMPRISES    STANDARD    WORKS 
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jr  ttuu  1  iticiice — noLBKOOK's  wormai  fliemoas — piOKTHJUNus  xeauuei  o  ^boi»iant 


THE  WORMAN  SERIES  IN  MODERN  LAN&UAGE. 

lOMPLETE  Course  in  German 

By  JAMES  H.  WORMAN,  A.M. 

.MKJSrrARY    GtERMLAl^ST    G-R-A^INlMAIi, 
COMPLETK    GERMAlSr    G-KAMMAIt, 

COLLiEOIAXltl    G-ERMIAT^    llJhZATi'E^Tl, 
KLEMEISTTARY    GJ-EFUVtiV^    READErj, 
IIVIAIS^    COPV-UOOKS,  GJ-ERnVlAlSr    ECHO. 

TORY    OF*    G-ERMLA.N"    IL.ITEP^A.TTJRE, 

OERlVIAlSr    JLN-D    ENGS^LISH    LEXICON-. 

n  i:  OERSfAy  GBAMMAJIS  of  Worman  are  widely  preferred  on  ac- 
■  their  clear,  explicit  method  (on  tlio  convei-uation  plan),  introclucing  a  eyptem 
./y  and  comparison  with  the  learuei-s'  own  language  and  oUiers  commonly 

.  of  understanding  the  spoken  language,  and  of  correct  pronun- 

1  great  tsuccess. 
..  »  .,,..-.-......,  .una  of  nouns  and  of  irregular  verbs  are  of  great  value  to  the 

The  use  of  heavy  type  to  indicate  etymological  changes,  is  new.  The  Vocabu- 
tfnonyiiiicai—VLlso  a  new  feature. 

»V> /.' >f  I  .V'S  GEItMAN  TtFJADJEn  contains  progrespive  Pelections 
vi  ,"  :  jf  of  the  very  bent  German  author;*,  including  thn-e  couipli-te  i>lay8, 
r     :  .  iuiuhased  in  eeparate  form  for  advanced  ttudents  who  have  com- 

■     rler. 
1'.  1  eminent  authors.  Notes  after  the  text.  References  to  all  Gler- 

i:  arnon  use,  and  SLC  ad^-iiyite  Vocabulary;  also.  Exercises  for 

.  iutu  thu  Uciinan. 

MOWirfV'.v  GETtHTAy^   ECTTO  (Tk^tfi^hfl*  Erhfi)    Is    entirely  a  new 

'  "     ••■  "  without  translation, 

;v. 

!i'  language.  By  this 
-[)raK>  II.  j-<<r  iin;  iiiiu'  uein-  no  i:^  u  iremian  tbroogn  and 
-  process  of  translating  his  thoughts  no  longer  Impedes  free 


IMAN'S  COMPLETE  FRENCH  COURSE 

18   INAUOLTIATED   BY 

ich  Echo;"  on  a  oLin  identical  with  the  German  Echo  described  above. 
This  will  be  foLowod  in  due  coarse  by  the  other  volumes  of 

•TECE   ERElSrCH   SERIES, 
VIZ.: 
^IJ'LETE  GRAMMAR,  [A    FRENCH    HEADER, 

T.  EMKSTA  RY  GRA  MM  A  7?,  i^     FRENCH    LEXICON, 
-t    HISTORY  OE  FRENCH  LITERATURE. 


WORMAN'S    WORKS 


\  nfl  fiist  as  published  by  many  of  the  best  institutions  of  the  country.    In 
•^s.  adaptation,  end  homogeneity  for  consistent  courses  of  instruction,  they 
are  simply 


A  WeU  of  EngUsh  Undefiled." 


LITERATURE  AND  BELLES  LETTRES. 


PROFESSOR  CLEVELAND'S  WORKS. 

A.  WHOLE  LIBRARY  IN  FOT7B,  VOLUMES. 


COlEffllUI^LITEMTIIBE. 

One  Hundred  and  Twenty  Thousand  of  these  Volumes  have  been  sold, 

and  they  are  the  acknowledged  Standard  wherever 

this  refining  study  is  pursued. 

PROP.  JAMES  R.  BOYD'S  WORKS. 

EMBEACING 

COMPOSITIOK,  LOGIC,   LITERATURE,  RHETORIC,  CRITICESM, 
BIOGRAPHY f— POETRY,  AND  PROSE, 


BOYD'S  COMPOSITION  AND  RHETORIC. 

ReuMirkable  for  the  space  and  attention  given  to  grammatical  principles,  to  afford  a 
substantial  groundwork ;  also  fw  the  admimble  treatment  of  synonyms,  figurative 
language,  and  the  sources  of  argument  and  illustration,  with  notable  exercises  for  pre- 
pariag  the  w&j  to  poetic  compositica. 

•     BOYD'S  ELEMENTS  OF  LOGIC. 

explains,  first,  the  conditions  and  processes  by  which  the  mind  receives  ideas,  and 
then  unfolds  the  art  of  reasoning,  with  clear  directions  for  the  establishment  and  con- 
firmation of  sound  judgment.  A  thoroughly  practical  treatise,  being  a  systematic  and 
philosophical  condensation  of  afl  that  is  known  of  the  subject. 

BOYD'S  KAMES'  CRITICISM. 

This  standard  work,  as  is  well  known,  treats  of  the  faculty  of  perception,  and  the 
result  of  its  exercise  upon  the  tastes  and  emotions.  It  may  therefore  be  termed  a  Com- 
pendium of  Aesthetics  and  Natural  Morals ;  and  its  use  in  refining  the  mind  and  heart 
has  made  it  a  standard  text-book. 

BOYD'S  ANNOTATED  ENGLISH  CLASSICS. 


Milton's  Paradise  Lost, 
^Young's  Night  TJioughts. 
Cowper's  Task,  Table  Talk,  &c. 


Tliontson's  Seasons, 
Pollok's  Course  of  Time. 
Lord  Bacon's  Essays. 


In  six  cheap  volumes.  The  service  done  to  literature,  by  Prof.  Boyd's  Annotations 
upon  these  standard  writers,  can  with  difficulty  be  estimated.  Line  by  line  their  ex- 
pressions and  ideas  are  analyzed  and  discussed,  until  the  best  comprehension  of  the 
powerful  use  of  language  is  obtained  by  the  learner. 


NEW 


ELEMENTARY  AlGEMA. 


BUBBACIKO 


THE  FIRST  PRINCIPLES  OF  THE  SCIENCE. 


BY 

CHARLES  DAVIES,  IJ.J>., 

PB0rBB6UK    or    BianiCS    SfATBEMATine,    COLUMBIA    COLLRGK. 


A.    S.    BARNES   &    C  0  M  T  A  X  Y, 
NEW  YORK  AND  CHICAaO. 

1875. 


IN  MEMORrAM  h/AI^^ 

DAVIES'  MATHEMATICS. 


THE    WEST     POINT     COURSE, 

And  Only  Thorough  and  Complete  Mathematical  Series. 


iisr    Tia:R,EE    f.a.i?,ts- 


. ;  .        I.;  a-OMMOi^r  school  oouese. 

I>avies'  Primary  iArcthmetic-  — The  fundamental  principles  displayed  in 

-  ,    Obj«>c-t  Lessone^       , 

IC^vi^'^  inte.U^ctTial'  Arithmetic— Referring  all  operations  to  the  unit  1  as 

the  only'tangible  liasls  for  logical  development. 
Savies'  Slements  of  Written  Arithmetic— A  practical  introduction  to 

the  whole  subject.    Theory  subordinated  to  Practice. 
Davies^  Practical  Arithmetic**— The  most  successful  combination  of  Theory 

and  Practice,  clear,  exact,  brief,  and  comprehensive. 

II.   ACADEMIC    COUESE. 

Davies'  University  Arithmetic*— Treating  the  subject  exhaustively  as 

a  science^  in  a  logical  series  of  connected  propositions. 
Davies'  Elementary  Alg-ebra-*— A  connecting  link,  conducting  the  pupil 

easUy  from  arithmetical  processes  to  abstract  analysis. 
Da  vies'  University  Algebra  ■*— For  institutions  desiring  a  more  complete 

but  not  the  fullest  course  in  pure  Algebra. 
Davies'  Practical  lYIathematics-— The  science  practically  applied  to  the 

useful  arts,  as  Drawing,  Architecture,  Surveying,  Mechanics,  etc. 
Davies'  Xilementary  CS-eometry .— The  important  principles  in  simple  form, 

but  with  all  the  exactness  of  vigorous  reasoning. 
Savies'  Elements  of  Surveying.- Re-written  in  1870.     The  simplest  and 

most  practical  presentation  for  youths  of  12  to  16. 

III.    COLLEGIATE    OOUESE. 

Davies'  Bourdon's  Algebra- *— Embracing  Sturm's   Theorem,  and  a  most 
exhaustive  and  scholarly  course. 

Davies'  University  Algebra-*— A  shorter  course  than  Bourdon,  for  Institu- 
tions have  less  time  to  give  the  subject. 

Davies'  Legendre's  Geometry-— Acknowledged  tJie  only  satisfactory  treatisa 
of  its  grade.    300,000  copies  have  been  sold. 

Davies'  Analytical  Geometry  and  Calculus-— The  shorter  treatises, 
combined  in  one  volume,  are  more  available  for  American  courses  of  study. 

Davies'  Analytical  Geometry- 1  The  original  compendiums,  for  those  de- 

Davies'  DifT-  &  Int-  Calculus-    '     siring  to  give  full  time  to  each  branch. 

Davies'  Descriptive  Geometry-— With  application  to  Spherical  Trigonome- 
try, Spherical  Projections,  and  Warped  Surfaces. 

Davies'  Shsdes,  Shadows,  and  Perspective-— A  succinct  expcsition  of 
the  mathematical  principles  involved. 

Davies'  Science  of  IKlathematics-- For  teachers,  embracing 

I.  Gbammar  of  Akithtietic,  ni.  Logic  and  Utility  of  Mathematics, 

II.  OuTxiNES  OF  Mathematics,  IV.  Mathematicax  Dictionakt. 


*  Keys  may  be  obtained  from  the  Publishers  by  Teachers  only. 

Entered,  according  to  Act  of  Congress,  in  the  year  1859,  ^y 

CHARLES     DAVIES, 

In  the  Qerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southern  District  of 

New  York, 
N.  E.  A. 


PREFACE. 


Algebra  naturally  follows  Arithmetic  in  a  course  of  eoieii- 
tific  studies.  The  language  of  figures,  and  the  elementary 
combinations  of  numbers,  are  acquired  at  an  early  age. 
When  the  pupil  passes  to  a  new  system,  conducted  by 
letters  and  signs,  the  change  seems  abrupt;  and  he  often 
experiences  much  difficulty  before  perceiving  that  Algebra 
is  but  Arithmetic  written  in  a  different  language. 

It  is  the  design  of  this  work  to  supply  a  connecting  link 
between  Arithmetic  and  Algebra ;  to  indicate  the  unity  of 
the  methods,  and  to  conduct  the  pupil  from  the  arithmetical 
processes  to  the  more  abstract  methods  of  analysis,  by  easy 
and  simple  gradations.  The  work  is  also  introductory  to 
the  University  Algebra,  and  to  the  Algebra  of  M.  Bourdon, 
which  is  justly  considered,  both  in  this  country  and  in 
Europe,  as  the  best  text-book  on  the  subject,  which  has  yet 
appeared. 

In  the  Introduction,  or  Mental  Exercises,  the  language 
of  figures  and  letters  are  both  employed.  Each  Lesson  is 
80  arranged  as  to  introduce  a  single  princi]>le,  not  known 

iii 

92G515 


IV  PREFACE. 

before,  and  the  whole  is  so  combined  as  to  prepare  the 
pupil,  by  a  thorough  system  of  mental  training,  for  those 
processes  of  reasoning  which  are  peculiar  to  the  algebraic 
analysis. 

It  is  about  twenty  years  since  the  first  publication  of  the 
Elementary  Algebra.  Within  that  time,  great  changes 
have  taken  place  in  the  schools  of  the  country.  Tlie  sys- 
tems of  mathematical  instruction  have  been  improved,  now 
methods  have  been  developed,  and  these  require  correspond- 
ing modifications  in  the  text-books.  Those  modifications 
have  now  been  made,  and  this  work  will  be  permanent  in 
its  present  form. 

Many  changes  have  been  made  in  the  present  edition,  at 
the  suggestion  of  teachers  who  have  used  the  work,  and 
favored  me  with  their  opinions,  both  of  its  defects  and 
merits.  I  take  this  opportunity  of  thanking  them  for  the 
valuable  aid  they  have  rendered  me.  The  criticisms  of 
those  engaged  in  the  daily  business  of  teaching  are  invalu- 
able to  an  author;  and  I  shall  feel  myself  under  special 
obligation  to  all  who  will  be  at  the  trouble  to  communicate 
to  me,  at  any  time,  such  changes,  either  in  methods  or  lan- 
guage, as  their  experience  may  point  out.  It  is  only  through 
the  cordial  co-operation  of  teachers  and  authors — by  joint 
labors  and  mutual  efforts — that  the  text-books  of  the  country 
can  be  brought  to  any  reasonable  degree  of  perfection 

A  Key  to  this  volume  has  been  prepared  for  the  use  of 
Teachers  only. 


CONTENTS. 


CHAPTER      I. 
DEFnnnoKS  Aim  kxtlanatoby  biohs. 

PAOB.-i 

Algebra — Defin-tions — Explanation  of  the  Signs 3S-41 

Examples  in  writing  Algebraic  expressions 41 

Interpretation  of  Algebraic  language 42 

CHAPTER    II. 

FUNDAMENTAL    OPERATIONS. 

Addition — Rule — Examples 43-60 

Subtraction — Rule — Examples — Remarks 50-66 

Multiplication — Monomials — Polynomials 66-6S 

Division — Monomials 63-68 

Signification  of  the  symbol  a* 68-70 

Division  of  Polynomials — Examples 71-76 

CHAPTER    III. 

USEFUL   FORMULAS.      FAOTOBINO,    ETO. 

Formulas  (1),  (2),  (3),  (4),  (5),  and  (6) 76-79 

Factoring 79-81 

Greatest  Common  Divisor 81-84 

lioast  Common  Multiple  .  .^ B4-87 

CHAPTER    IV 

FRAOnONS. 

Transformation  of  Fractions , , ,  89 

To  Reduce  an  Entire  Quantity  to  a  Fractional  Form 90 

V 


VI  CONTENTS. 

To  Reduce  a  Fraction  to  its  Lowest  Terms 90 

To  Reduce  a  Fraction  to  a  Mixed  Quantity ■.  92 

To  Reduce  a  Mixed  Quantity  to  a  Fraction 93 

To  Reduce  Fractions  to  a  Common  Denominator »  94 

Addition  of  Fractions 96 -9c 

Subtraction  of  Fractions 98-  ?9 

Multiplication  of  Fractions ,  99-  10^ 

Division  of  Fractions 102-  1 05 

CHAPTER    y. 

EQUATIONS  OF  THE  FIRST   DEGREK. 

Definition  of  an  Equation — Different  Kinds 106-106 

Transformation  of  Equations — First  and  Second 106-110 

Solution  of  Equations — Rule 110-114 

Problems  involving  Equations  of  the  First  Degree 115-130 

Equations  involving  Two  Unknown  Quantities 130-131 

Elimination — By  Addition — By  Subtraction — By  Comparison. .  131-143 

Problems  involving  Two  Unknown  Quantities , 143-148 

Equations  involring  Three  or  more  Unknown  Quantities 148-159 

CHAPTER    VI. 

FOBMATION   OF  P0WEK8. 

Definition  of  Powers 160-161 

Powers  of  Monomials 161-163 

Powers  of  Fractions 163-165 

Powers  of  Biuomials 165-167 

Of  the  Terms— Exponents— Coefficients 167-170 

Binomial  Formula — Examples 170-172 

CHAPTER    VII. 

SQUARE  BOOT.      RADICALS   OF  THE  SECOND   DEGREE. 

Definition— Perfect  Squares— Rule— Examples 173-179 

Square  Root  of  Fractious 179-181 

Square  Root  of  Monomials 181-183 

Imperfect  Squares,  or  Radicals 1 83-1 87 

Addition  of  Radicals 187-189 

Subtraction  of  Radicals 189-190 


0  ()  N  T  K  N  T  S  .  Vii 

rAGES. 

Multiplication  of  Radicals 190-191 

Division  of  Kadicals 191-192 

Square  Root  of  Polynomials 193-197 

CHAPTER    VIII. 

ZQUATIONS    or  TUB  SECOND   DEORXX. 

Eiiuations  of  the  Second  Degree — Definition — Form 198-200 

Incomplete  Equations 200-209 

Complete  Equations— Rule 209-211 

Four  Forms 211-227 

Four  Properties 227-229 

Formation  of  Equations  of  the  Second  Degree 229-231 

Numerical  Values  of  the  Roots 281-236 

Problems 236-240 

Equations  involving  more  than  One  Unknown  Quantity 241-250 

Problems 260-264 

CHAPTER    IX. 

ARrrnxETicAL  and  oeometrioal  rRopoimoN. 

Ways  in  which  Two  Quantities  may  be  Compared 256 

Arithmetical  Proportion  and  Progression 266-267 

Last  Term 267-260 

Sum  of  the  Extremes— Sum  of  Series 260-262 

The  Five  Numbers — To  find  any  number  of  Means 262-265 

Geometrical  Proportion 267 

Various  Kinds  of  Proportion 268-278 

Geometrical  Progression 278-280 

Last  Term-  Sum  of  Scries 280-286 

Progression  having  an  Infinity  Number  of  Terms 286-288 

Tho  live  Numbers— To  find  One  Mean 288-289 

CHAPTER    X. 

LooABrraus. 

Tlieory  of  Logarithms 290-295 


SUGGESTIONS    TO    TEACHERS. 


1.  The  Introduction  is  designed  as  a  mental  exercise.  If 
thoroughly  taught,  it  will  train  and  prepare  the  mind  of 
the  pupil  for  those  higher  processes  of  reasoning,  which  it 
is  the  peculiar  province  of  the  algebraic  analysis  to  develop. 

2.  The  statement  of  each  question  should  be  made,  and 
every  step  in  the  solution  gone  through  with,  without  the 
aid  of  a  slate  or  black-board  ;  though  perhaps,  in  the  begin- 
ning, some  aid  may  be  necessary  to  those  unaccustomed  to 
such  exercises. 

3.  Great  care  must  be  taken  to  have  every  principle  on 
which  the  statement  depends,  carefully  analyzed ;  and  equal 
care  is  necessary  to  have  every  step  in  the  solution  distinctly 
explained. 

4.  The  reasoning  process  is  the  logical  connection  of  dis- 
tinct apprehensions,  and  the  deduction  of  the  consequences 
which  follow  from  such  a  connection.  Hence,  the  basis  of 
all  reasoning  must  lie  in  distinct  elementary  ideas. 

5.  Therefore,  to  teach  one  thing  at  a  time — to  teach  that 
thing  well — to  explain  its  connections  vnth  other  thmgs 
and  the  consequences  which  follow  from  such  connections, 
would  seem  to  embrace  the  whole  art  of  instruction. 

viii 


ELEMENTAaY   ALGEBRA, 


INTKODUCIION. 

MENTAL      EXERCISES. 

LESSON    L 

1.  John  and  Charles  have  the  same  number  of  apples; 
both  together  have  twelve :  how  many  has  each  ? 

Analysis. — Let  x  denote  the  number  which  John  has; 
then,  since  they  have  an  equal  number,  x  will  also  denote 
the  number  which  Charles  has,  and  t^vice  x,  or  2ic,  will 
denote  the  number  which  both  have,  which  is  12.  If  twice 
x  is  equal  to  12,  x  will  be  equal  to  12  divided  by  2,  <vhich 
ifi  6  ;  therefore,  each  has  6  apples. 

WRriTEN. 

Let  X  denote  the  number  of  apples  which  John  }ia«; 
then, 

12 
X  -\-  X  =  2x  =  12;    hence,    a;  =  —  =  6. 

Note. — ^When  x  is  written  with  the  sign  +  before  it, 
it  is  read  plus  x :  and  the  line  above,  is  read,  x  plus  x 
rqiuils    12. 


10  1  N  T  E  O  D  U  C  T  I  0  N  . 

S"(>TE. — Wiien    a-    18  written  by  itself,  it  is  read  one  a5j 
^anfl  i^;tiie.sf?-iiic  as,   la* ; 

"  *    '   X   or    Ice,  means  once              cc,  or  one  a^ 

2a;,  "       twice            a;,  or  two  a;, 

3ic,  "       three  times  a*,  or  three  a;, 

4a;,  "       four  times    a;,  or  four  x, 

&c.,  &c.,                    &c. 

2.  What  is  a;  +  a;  equal  to  ?  " 

3.  What  is  a;  +  2a; .  equal  to  ? 

4.  What  is  a;  +  2a;  +    x   equal  to  ? 

5.  What  is  a;  -f  5a;  4-    x   equal  to  ? 

6.  What  is  a;  +  2a:  +  3a;  equal  to  ? 

7.  James  and  John  together  have  twenty-four  peaclies, 
and  one  has  as  many  as  the  other ;  how  many  has  each  ? 

Analysis. — Let  x  denote  the  number  which  James  has ; 
then,  since  they  have  an  equal  number,  x  will  also  denote 
the  number  which  John  has,  and  twice  x  will  denote  the 
number  which  both  have,  which  is  24.  If  twice  x  is  equal 
to  24,  X  will  be  equal  to  24  divided  by  2,  which  is  12  ; 
therefore,  each  has  12  peaches. 

WRITTEN. 

Let  X  denote  the  number  of  peaches  which  James  has ; 
then, 

24 
a;  +  a;,  =r  2a;  =  24;    hence,    a;  —  —   =12. 

VERIFICATION. 

.    A  l^erification  is  the  operation  of  proving  that  the  num. 
ber  found  mil  satisfy  the  conditions  of  the  question.     Thus, 

James'  appiCS.  John's  apples. 

12  4-12  =   24. 

Note. — Let  the  following  questions  be  analyzed^  written^ 
and  verified^  in  exactly  the  sa^ne  manner  as  the  above. 


MENTAL      EXERCISES.       .  11 

8.  William  and  John  together  have  36  pears,  and  one  has 
as  many  as  the  other  :  how  many  has  each  ? 

9.  What  number  added  to  itself  will  make  20? 

10.  James  and  John  are  of  the  same  age,  and  the  sum  of 
their  ages  is  32  :  what  is  the  age  of  each? 

11.  Lucy  and  Ann  are  twins,  and  the  sum  of  their  ages 
Is  10:  M'hat  is  the  age  of  each? 

12.  What  number  is  that  Avhich  added  to  itself  will 
make  30? 

13.  What  number  is  that  which  added  to  itself  will 
make  50? 

14.  Each  of  two  boys  received  an  equal  sum  of  money  at 
Chiistmas,  and  together  they  received  60  cents:  how  much 
had  each  ? 

15.  What  number  added  to  itself -vvdll  make  100? 

16.  John  has  as  many  pears  as  William;  together  they 
have  72  :  how  many  has  each  ? 

1 7.  What  number  added  to  itself  wHl  give  a  sum  equal 
to  46  ? 

1 8.  Lucy  and  Ann  have  each  a  rose  bush  with  the  same 
number  of  buds  on  each ;  the  buds  on  both  number  46 : 
how  many  on  each? 


LESSON  n. 

1.  John  and  Charles  together  have  12  apples,  and  Charles 
has  twipe  as  many  as  John :  how  many  has  each  ? 

Analysis. — Let  x  denote  the  number  of  apples  which 
John  has ;  then,  since  Charles  has  twice  as  many,  2x  wijl 
denote  his  share,  and  a;  4-  2aj,  or  3sc,  will  denote  the 
number  which  they  both  have,  which  is  12.^  If  3a!  is  equal 
lo  12,  X  will  be  equal  to  12  divided  by  3,  which  is  4; 
therefore,  John  has  4  apples,  and  Charles,  having  tv\nce  as 
many,  has  8. 


12  ^  »N'I  RODUCTION. 

writt:^t. 
TiCt  X  denote  the  number  of  apples  John  has ,  then, 
2x  will  denote  the  number  of  apples  Charles  has;  and 
X  -t  2x  =  3x  =z  12,    the  number  both  have;  then, 

12 
X  =  -~   =     4,    the  number  John  has ;  and, 

2a;  =  2  X  4  =  8,    the  number  Charles  has. 

,  VERIFICATION. 

4  +  8  =   12,    the  number  both  have. 

2.  William  and  John  together  have  48  quills,  and,  Williara 
has  twice  as  many  as  John :  how  many  has  each  ? 

3.  What  number  is  that  which  added  to  twice  itselfj  will 
give  a  number  equal  to  60  ? 

4.  Charles'  marbles  added  to  John's  make  3  times  as  many 
as  Charles  has;  together  they  have 51 :  how  many  has  each  ? 

AxALTSis. — Since  Charles'  marbles  added  to  John's  make 
three  times  as  many  as  Charles  has,  Charles  must  have  one 
third,  and  John  two  thirds  of  the  whole. 

Let  X  denote  the  number  which  Charles  has ;  then  2x 
will  denote  the  number  which  John  has,  and  x  -\-  2cc,  or 
3x,  will  denote  what  they  both  have,  which  is  51.  Then,  if 
3x  is  equal  to  51,  x  will  be  equal  to  51  divided  by  3, 
which  is  17.  Therefore,  Charles  has  17  marbles,  and  John, 
having  twice  as  many,  has  34. 

WKITTEN. 

Let   X   denote  the  number  of  Charles'  marbles;  then, 
2x  will  denote  the  number  of  John's  marbles;  and 

'     Sx  =  51,    the  number  of  both;  then, 

51 
a;  —  —  =  17,    Charles'  marbles;  and 

17  X  2   =   34,    Jolm's  marbles. 


MENTAL      BZESCI8E6.  13 

5.  What  number  added  to  twice  itself  will  make  15  ? 

6.  AVhat  number  added  to  t\\nce  itself  will  make  57  ? 

7.  What  number  added  to  twice  itself  will  make  39? 

8.  What  number  added  to  twice  itseff  will  give  90  ? 

9.  John  walks  a  certain  distance  on  Tuesday,  twice  20* 
for  on  Wednesday,  and  in  the  two  days  he  walks  27  miles: 
how  far  did  he  walk  each  day  ? 

10.  Jane's  bush  has  twice  as  many  roses  as  Nancy's:  and 
on  both  bushes  there  are  36 :  how  many  on  each  ? 

11.  Samuel  and  James  bought  a  ball  for  48  cents ;  Samuel 
paid  twice  as  much  as  James :  what  did  each  pay  ? 

12.  Divide  48  into  two  such  parts  that  one  shall  be  double 
the  other. 

13.  Divide  60  into  two  such  parts  that  one  shall  be  double 
the  other, 

14.  The  sum  of  three  equal  numbers  is  12  :  what  are  the 
numbers  ? 

Analysis. — Let  x  denote  one  of  the  numbers;  then, 
since  the  numbers  are  equal,  x  \snll  also  denote  each  of 
the  others,  and  x  plus  x  plus  x,  or  3a!  will  denote  their 
sum,  which  is  12.  Then,  if  3aj  is  equal  to  12,  x  will  be 
equal  to  12  divided  by  3,  which  is  4 :  therefore,  the  numbers 
are  4,  4,  and  4. 

WRriTEN. 

Let  x  denote   one  of  the  equal  numbers ;  then, 
x  +  X  -h  X  =  Sx  =  12;    and 

x  =  -  =     4. 

VERIFICATION. 

4  -f  4  +  4   =   12. 

15.  Tlie  sum  of  three  equal  numbers  is  24 :  what  are  the 
numbers? 

16.  The  sum  of  three  equal  numbers  is  36  :  what  are  the 
numbers? 


14  I  K  T  K  O  U  U  C  T  I O  K  . 

17.  The  sum  of  three  equal  numbers  is  54  :  what  are  the 
numbers  ? 


LESSON    III. 

1.  What  number  is  that  which  added  to  three  times  itself 
will  make  48  r 

Analysis. — Let  x  denote  the  number;  then,  3a!  will 
denote  three  times  the  number,  and  x  plus  3£c,  or  A^x, 
will  denote  the  sum,  which  is  48.  If  Ax  is  equal  to  48, 
X  will  be  equal  to  48  divided  by  4,  which  is  12;  there- 
fore,  12  is  the  required  number. 

WEITTEN. 

Let   X   denote  the  number;    then, 

'dx  —  tln-ee  times  the  number;    and 
ic  +  3a;  =   4a;  :=  48,    the  sum :    then, 

a;  1=  —  =   12,    the  required  number. 

VEllIFICATION. 

12  +  3  X   12   r=   12  +  38    =:  48. 

Note. — All  similar  questions  are  solved  by  the  same 
form  of  analysis. 

2.  What  number  added  to  4  times  itself  will  give  40  ?  ^ 

3.  What  number  added  to  5  times  itself  will  give  42  ?  *! 

4.  What  number  added  to  6  times  itself  will  give  63?  ^ 
6.  What  number  added  to  7  times  itself  will  give  88  ? 

6.  What  number  added  to  8  times  itself  will  give  81  ? 

7.  What  number  added  to  9  times  itself  will  give  100? 

8.  James  and  John  together  have  24  quills,  and  John  has 
three  times  as  many  as  James:  how  many  has  each? 

9.  William  and  Charles  have  64  marbles,  and  Charles  has 
7  times  as  many  as  William :  how  many  has  each  ? 


MENTAL      KXERCI6E6.  15 

10.  James  and  John  travel  96  miles,  and  James  travels 
1 1  times  as  far  as  Jolin  :  how  far  does  each  travel  ? 

11.  The  sum  of  the  ages  of  a  father  and  son  is  84  years; 
and  the  father  is  3  times  as  old  as»  the  son :  what  is  the  aire 
of  each? 

12.  There  are  two  numbers  of  which  the  greater  is  1 
tmies  the  less,  and  their  sum  is  72  :  what  are  the  nmnbers? 

13.  The  sum  of  four  equal  numbers  is  64:  what  are  the 
numbere  ? 

14.  The  sum  of  six  equal  numbers  is  54;  what  are  the 
numbers  ? 

15.  James  has  24  marbles ;  he  loses  a  certain  number,  and 
then  gives  away  7  times  as  many  as  he  loses  which  takes  all 
he  has :  how  many  did  he  give  away  ?     Verify. 

16.  William  has  36  cents,  and  divides  them  between  his 
two  brothers,  James  and  Charles,  giving  one,  eight  times  as 
many  as  the  other :  how  many  does  he  give  to  each  ? 

17.  What  is  the  sum  of  x  and  3a;?  Of  a;  and  7aj? 
Of  X  and  5a;?     Of  x  and   12a;? 


LESSON  rv. 

1.  If  1  apple  costs  lucent,  what  will  a  number  of  apples 
denoted  by  x  cost:? 

Analysis. — Since  one  apple  costs  1  cent,  and  since  x 
denotes  a?iy  number  of  apples,  the  cost  of  x  apples  will  be 
as  many  cents  as  there  are  apples :  that  is,  x  cents. 

2.  If  1  apple  costs  2  cents,  what  will  x  apples  cost? 

Analysis. — Since  one  apple  costs  2  cents,  and  smce  a 
denotes  the  number  of  a})ples,  the  cost  will  be  twice  as  many 
cents  as  there  are  apples :  that  is  2a;  cents. 

3.  If  1  apple  costs  3  cents,  what  H-ill  x  apples  cost  ? 

4.  If  1  lemon  costs  4  cents,  wliat  will  x  lemons  cost? 


16  INTRODUCTION. 

5.  K  1  orange  costs  6  cents,  what  will  x  oranges  cost  ? 

6.  Charles  bought  a  certain  number  of  lemons  at  2  cents 
apiece,  and  as  many  .oranges  at  3  cents  apiece,  and  paid  in  all 
20  cents :  how  many  did  he  buy  of  each  ? 

Analysis. — ^Let  x  denote  the  number  of  lemons ;  then, 
since  he  bought  as  many  oranges  as  lemons,  :t  will  also 
denote  the  number  of  oranges.  Since  the  lemons  were 
2  cents  apiece,  2x  will  denote  the  cost  of  the  lemons ;  and 
since  the  oranges  were  3  cents  apiece,  3a;  will  denote 
the  cost  of  the  oranges ;  and  2a5  +  3a;,  or  6aj,  "vvill  denote 
the  cost  of  both,  which  is  20  cents.  Now,  since  hx  cents 
are  equal  to  20  cents,  x  will  be  equal  to  20  cents  divided  by 
5  cents,  which  is  4 :  hence,  he  bought  4  of  each. 

WEITTEN. 

Let  X  denote  the  number  of  lemons,  or  oranges ;  then, 
2a;  —  the  cost  of  the  lemons ;  and 
3a;  =  the  cost  of  the  oranges ;  hence, 
2a;  -j-  3a;  =  5a;  —  20  cents  =  the    cost    of   lemons   and 

oranges;  hence, 

X  = =  4,  the  number  of  each. 

5  cents 

VERIFICATION. 

4  lemons  at  2  cents  each,  give,  4x2=     8  cents. 

4  oranges  at  3  cents  each,     "     4  x  3   =   12  cents. 

Hence,  they  both  cost,  8  cents  +12  cents  =  20  cents. 

v.  A  farmer  bought  a  certain  number  of  sheep  at  4  dollars 
apiece,  and  an  equal  number  of  lambs  at  1  dollar  a])iece, 
and  the  whole  cost  60  dollars:  hew  many  did  he  buy  of 
each  ? 

8.  Charles  bought  a  certain  number  of  apples  at  1  cent 
apiece,  and  an  equal  number  of  oranges  at  4  cents  apiece,  and 
paid  50  cents  in  all :  how  many  did  he  buy  of  each  ? 


MENTAL      EXERCISES.  17 

9.  James  bought  an  equal  number  of  apples,  pears,  and 
lemons ;  he  paid  1  cent  apiece  for  the  apples,  2  cents  apiece 
for  the  pears,  and  3  cents  apiece  for  the  lemons,  and  paid 
12  cents  in  all :  how  many  did  he  buy  of  each  ?     Verify. 

10.  A  farmer  bought  an  equal  number  of  sheep,  hogs, 
and  calves,  for  which  he  paid  108  dollars;  he  paid  3  dollars 
apiece  for  the  sheep,  5  dollars  apiece  for  the  hogs,  and 
4  dollars  apiece  for  the  calves :  how  many  did  he  buy  of 
each? 

11.  A  farmer  sold  an  equal  number  of  ducks,  geese, 
and  turkeys,  for  which  he  received  90  shillings.  The  ducks 
brought  Mm  3  shillings  apiece,  the  geese  6,  and  the  tui-keys 
1 :  how  many  did  he  sell  of  each  sort  ? 

12.  A  tailor  bought,  for  one  hundred  dollars,  two  pieces 
of  cloth,  each  of  which  contained  an  equal  number  of  yards. 
For  one  piece  he  paid  3  dollars  a  yard,  and  for  the  other 
2  dollars  a  yard ;  how  many  yards  in  each  piece  ? 

13.  The  sum  of  three  numbers  is  28  ;  the  second  is  tmce 
the  first,  and  the  third  twice  the  second :  Avhat  are  the 
numbers  ?     Verily. 

14.  The  sum  of  three  numbers  is  64  ;  the  second  is  3  times 
the  fipt,  and  the  third  4  tunes  the  second :  what  are  the 
numbers  ? 


LESSON    V. 

1.  If  1  yard  of  cloth  costs  x  dollars,  what  will  2  yards 
cost? 

Analysis. — ^Two  yards  of  cloth  will  cost  twice  as  much  as 
one  yard.  Therefore,  if  1  yard  of  cloth  costs  x  dollars, 
2  yards  will  cost  twice  x  dollars,  or  2x  dollars. 

2.  If  1  yard  of  cloth  costs  x  dollars,  what  will  3  yards 
cost?    Why? 


18  INTKODUCTION. 

3.  If  1  orange  costs  x  cents,  what  will  1  oranges  cost  \ 
Why  ?     8  oranges  ? 

4.  Charles  bought  3  lemons  and  4  oranges,  for  which  he 
paid  22  cents.  He  paid  twice  as  much  for  an  orange  as  for 
a  lemon  :  what  was  the  price  of  each  ? 

An-alysis. — Let  x  denote  the  price  of  a  lemon ;  then,  2i8 
will  denote  the  price  of  an  orange ;  3a5  wdll  denote  the  cost 
of  3  lemons,  and  Sec  the  cost  of  4  oranges ;  hence,  Zx  plus 
8a?,  or  11  ic,  will  denote  the  cost  of  the  lemons  and  oranges, 
which  is  22  cents.  If  \\x  is  equal  to  22  cents,  x  is  equal  to 
22  cents  divided  by  11,  which  is  2  cents:  therefore,  the 
price  of  1  lemon  is  2  cents,  and  that  of  1  orange  4  cents. 

AVKITTEN. 

Let  X  denote  the  price  of  1  lemon ;  then, 
2a;  =  "  1  orange ;  and, 

3rr -f  8a;  =  11a;  r=  22  cts.,  the  cost  of  lemons  and  oranges; 

22  cts 
hence,  x  =  — — — '-  —  2  cts.,  the  price  of  1  lemon ; 

and,  2x2  =  4  cts.,  the  price  of  1  orange;. 

VERIFICATION. 

3  X  2   =z     6  cents,  cost  of  lemons, 

4  X  4  =   16  cents,  cost  of  oranges. 

22  cents,  total  cost. 

6.  James  bought  8  apples  and  3  oranges,  for  which  he 
paid  20  cents.  He  paid  as  much  for  1  orange  as  for  4  applss-* 
what  did  he  pay  for  one  of  each  ? 

6  A  farmer  bought  3  calves  and  7  pigs,  for  which  he  paid 
19  dollars.  He  paid  four  times  as  much  for  a  calf  as  for  a 
pig :  what  was  the  price  of  each  ? 

*l.  James  bought  an  apple,  a  peach,  and  a  pear,  for  which 
ho  paid  6  cents.     He  paid  twice  as  much  for  the  peach  as  for 


MENTAL      EXER0I6EB.  19 

the  apple,  and  three  tunes  as  much  for  the  pear  as  for  the 
apple :  what  was  the  j)rice  of  each  ? 

8.  Williaiu  bought  an  ai)ple,  a  lemon,  and  an  orange,  for 
wliich  lie  paid  24  cents,  lie  paid  twace  as  much  for  the 
lemon  as  for  the  apple,  and  3  times  as  muck  for  the  orange 
as  for  the  apple :  what  was  the  j)rice  of  each  ? 

9.  A  fanuer  sold  4  calves  and  5  cows,  for  which  he  received 
120  dollars.  He  received  as  much  for  1  cow  as  for  4  calves: 
what  was  the  price  of  each  ? 

10.  Lucy  bought  3  ])ears  and  5  oranges,  for  which  she 
piud  20  cents,  giving  t^^'ice  as  much  for  each  orange  as  for 
jach  pear:  what  was  the  price  of  each? 

11.  Ann  bought  2  skeins  of  silk,  3  pieces  of  tape,  and  a 
penknife,  for  which  she  paid  80  cents.  She  paid  the  same 
for  the  silk  as  for  the  tape,  and  as  much  for  the  penknife  as 
for  both  :  what  was  the  cost  of  each  ? 

12.  James,  John,  and  Charles  are  to  divide  66  cents 
among^  them,  so  that  John  shall  have  twice  as  many  as 
James,  and  Charles  twice  as  many  as  John:  what  is  the 
share  of  each  ? 

13.  Put  54  aj)ple8  into  three  baskets,  so  that  the  second 
hall  contain  t^Wce  as  many  as  the  first,  and  the  third  as 
1  any  as  the  first  and  second:  how  many  will  there  be  hi 
ich. 

14.  Divide  60  into  four  such  parts  that  the  second  shaU 
be  double  the  first,  the  third  double  the  second,  and  the 
Iburth  double  the  third :  what  are  tlie  numbers  ? 


LESSON    VL 

1.  If   2x  +  X    is  equal  to    Sx,    what  is    3x  —  x    eqiuil 
to  ?     Written,  3x  —  x  ==   2x. 

2.  What  is    4a;  —  CB    equal  to  ?    Written, 

4in  —  jc  —   3x. 


20  IN  TKOD  UOTION. 

3.  Wliat  is  8a3  minus  6a!  equal  to  ?     Written, 

8a;  —  6a;  =  2x. 

4.  What  is    12a;  —  9a;    equal  to?  Ans,  Sa. 
6.  What  is    15a;  —  7a;    equal  to? 

6.  What  is   •I'Jx  —  13a;    equal  to?  Arts,  4a;. 

V.  Two  men,  who  are  30  miles  apart,  travel  towards  each 
other  ;  one  at  the  rate  of  2  miles  an  hour,  and  the  other  at 
the  rate  of  3  miles  an  hour :  how  long  before  they  will  meet? 

Analysis. — Let  x  denote  the  number  of  hours.  Then, 
since  the  time,  multiplied  by  the  rate,  will  give  the  distance, 
1x  will  denote  the  distance  traveled  by  the  first,  and  3a! 
the  distance  traveled  by  the  second.  But  the  sum  of  the 
distances  is  30  miles ;   hence, 

2a;  4-  3a;  =  6a;  =  30  miles; 
and  if  bx  is  equal  to  30,  x  is  equal  to  30  divided  by  6, 
which  is  6  :  ^ence,  they  will  meet  in  6  hours. 

WRITTEN. 

Let  X  denote  the  time  in  hours ;  then, 

2a;  =  the  distance  traveled  by  the  1st ;  and 
3a;  =  i*  "  2d. 

By  the  conditions, 

2a;  4-  3a;  =  5a;  =  30  miles,  the  distance  apart ; 

hence,  jc  =  —  =  6  hours. 

o 

VERIFICATION. 

2x6  z=z  12  miles,  distance  traveled  by  the  first. 
3x6  =  18  miles,  distance  traveled  by  the  second 
30  miles,  whole  distance. 
8,  Two  persons  are  10  miles  apart,  and  are  traveling  in 
the  same  direction ;  the  first  at  the  rate  of  3  miles  an  hour, 
and  the  second  at  the  rate  of  5  miles :  how  long,  before  the 
second  will  overtake  the  first  ? 


and, 

5x 

= 

a 

and, 

5x 

— 

3x 

— 

2x 

or, 

X 

= 

10 
2 

MENTAL      EXEB0ISK8.  21 

Analysis. — Let  x  denote  the  time,  in  hours.  Then,  Sx 
will  denote  the  distance  traveled  by  the  first  in  x  hours; 
and  5x  the  distance  traveled  by  the  second.  But  when 
the  second  overtakes  the  first,  he  will  have  traveled  10  miles 
more  than  the  first :  hence, 

5aj  —  3a;  =  2a;  =   10; 
if  2a;  is  equal  to  10,  x  is  equal  to  5  •  hence,  the  second  will 
overtake  the  first  in  5  hours. 

WRITTEN. 

Let  X  denote  the  time,  in  hours:  then, 
3a;  =  the  distance  traveled  by  the  1st; 

2d; 
=   1 0  hours ; 

=     5  hours.  ^ 

VKRIFICATIQN. 

8x5  =  15  miles,  distance  traveled  by  1st. 
6  X  5   =  25  mUes,  "  "  2d. 

25  —  15  =  10  miles,  distance  apart. 

9.  A  cistern,  holding  100  hogsheads,  is  filled  by  two 
pipes ;  one  discharges  8  hogsheads  a  minute,  and  the  other 
12  :  in  what  time  will  they  fill  the  cistern? 

10.  A  cistern,  holding  120  hogsheads,  is  filled  by  3  pipes; 
tlie  first  discharges  4  hogsheads  in  a  minute,  the  second  7, 
and  the  third  1 :  in  what  time  will  they  fill  the  cistern  ? 

11.  A  cistern  which  holds  90  hogsheads,  is  filled  by  a  pipe 
^vhich  discharges  10  hogsheads  a  minute ;  but  there  is  a 
waste  pipe  which  loses  4  hogsheads  a  minute :  how  long 
will  it  take  to  fill  the  cistern  ? 

12.  Two  pieces  of  cloth  contain  each  an  equal  number  of 
yards ;  the  first  cost  3  dollars  a  yard,  and  the  second  5,  and 
both  pieces  cost  96  dollars :  how  many  yards  in  each  ? 

1 3.  Two  pieces  of  cloth  contain  each  an  equal  number  of 
yards ;  the  first  cost  7  dollars  a  yard,  and  the  second  6 ;  the  first 


22  INTRODUCTION. 

cost  60  dollars  more  than  the  second :  how  many  yards  m 
each  piece  ? 

14.  John  bought  an  equal  number  of  oranges  and  lemons 
the  oranges  cost  him  5  cents  apiece,  and  the  lemons  3 ;  and 
he  paid  56  cents  for  the  whole:  how  many  did  he  buy  of 
each  kind  ? 

15.  Charles  bought  an  equal  number  of  oranges  and 
lemons;  the  oranges  cost  him  5  cents  apiece,  and  the 
lemons  3  ;  he  paid  14  cents  more  for  the  oranges  than  for 
the  lemons :  how  many  did  he  buy  of  each  ? 

16.  Two  men  work  the  same  number  of  days,  the  one 
receives  1  dollar  a  day,  and  the  other  two :  at  the  end  of 
the  time  they  receive  54  dollars  :  how  long  did  they  work  ? 


LESSON^    yii.  ^ 

1.  John  and  Charles  together  have  25  cents,  and  Charles 
has  5  more  than  Jolm :  how  many  has  each  ? 

Analysis. — Let  x  denote  the  number  which  John  has  ; 
then,  a;  +  5  will  denote  the  number  which  Charles  has,  and 
a;  -h  85  +  5,  or  2a;  4-  5,  will  be  equal  to  25,  the  number 
they  both  have.  Since  2x  +  5  equals  25,  2x  will  be 
equal  to  25  minus  5,  or  20,  and  x  will  be  equal  to  20 
divided  by  2,  or  10:  therefore,  John  has  10  cents,  and 
Charles  15. 

WRITTEN. 

Let  X  denote  the  niunber  of  John's  cents ;  then, 
^  a;  -h  5   =r  "  Charles'  cents;  and, 

jc  -f  a;  -h  5  =.  25,  the  number  they  both  have ;  or, 
205  -J-  5  ^  25: ;     and, 

2a;  =  25—5   ==  20;     hence, 

20 
X  ■=.  —  =10,  John's  number;  and, 

z 
10  +  5   =   15,  Charles'  number. 


MENTAL 

BXKK0I6B6. 

VEBIPICATION. 

10 

Chwlea'. 
+    15 

=  25,    the  sum. 

ChM-le*' 
15 

John's. 
-    10 

=     5,    the  difference. 

23 


2.  James  and  John  have  30  marbles,  and  John  has  4  more 
than  James :  how  many  has  each,? 

3.  AVilHam  bought  GO  oranges  and  lemons ;  there  were 
20  more  lemons  than  oranges:  how  many  were  there  of 
each  sort  ? 

4.  A  farmer  has  20  more  cows  than  calves ;  in  all  he  has 
86  :  how  many  of  each  sort? 

5.  Lucy  has  28  pieces  of  money  in  her  purse,  composed 
of  cents  and  dimes;  the  cents  exceed  the  dimes  in  number 
by  16  :  how  many  are  there  of  each  sort  ? 

0.  Wliat  number  added  to  itself,  and  to  9,  will  make  29  ? 
Y.  What  number  added  to  twice  itself,  and  to  4,  will 
make  25  ? 

8.  What  number  added  to  three  tunes  itself,  and  to  12, 
will  make  60  ? 

9.  John  has  five  times  as  many  marbles  as  Charles,  and 
*'liat  they  both  have,  added  to  14,  makes  44 :  how  many  has 
tach? 

10.  There  are  three  numbers,  of  which  the  second  is  t^vice 
the  first,  and  the  third  twice  the  second,  and  when  9  is 
added  to  the  sum,  the  result  is  30 :  what  are  the  numbers? 

11.  Divide  17  into  two  such  parts  that  the  second  shall 
be  two  more  than  double  the  first:  what  are  the  parts? 

12.  Divide  40  Jnto  three  such  parts  that  the  second  shall 
be  twice  the  first,  and  the  third  exceed  six  times  the  first 
by  4  :  what  are  the  parts? 

13.  Charles  has  twice  as  many  cents  as  James,  and  John 


24  INTKODUCTION. 

has  twice  as  many  as  Charles ;  if  7  be  added  to  what  they 
all  have,  the  sum  will  be  28 :  how  many  has  each  ? 

14.  Divide  15  into  three  such  parts  that  the  second  shaU 
be  3  times  the  first,  the  third  twice  the  second,  and  5  over ; 
what  are  the  numbers  ? 

15.  An  orchard  contains  three  kinds  of  trees,  apples,  pears, 
and  cherries;  there  are  4  times  as  many  pears  as  apples, 
twice  as  many  cherries  as  pears,  and  if  14  be  added,  th«^ 
number  will  be  40 ;  how  many  are  there  of  each  ? 


LESSOR  vm. 

1.  John  after  giving  away  5  marbles,  had  12  left:  how 
many  had  he  at  first  ? 

Analysis. — Let  x  denote  the  number ;  then,  x  minus  5 
will  denote  what  he  had  left,  which  was  equal  to  12.  Since 
X  diminished  by  5  is  equal  to  12,  x  will  be  equal  to  12, 
increased  by  5  ;  that  is,  to  17 :  therefore,  he  had  17  marbles. 

WRITTEN. 

Let  X  denote  the  number  he  had  at  first;  then, 
05  —  5   =   12,    what  he  had  left;  and 

X  =  12  +  5  =  17,    what  he  first  had. 

VERIFICATION. 

17  —  5  =  12,    what  were  left. 

2.  Charles  lost  6  marbles  and  has  9  left :  how  many  had 
he  at  first  ? 

3.  \yilliam  gave  15  cents  to  John,  and  had  9  left:  how 
many  had  he  at  first  ? 

4.  Ann  plucked  8  buds  from  her  rose  bush,  and  there 
svere  19  left :  how  many  were  there  at  first  ? 


MENTAL       EXERCISES.  25 

6.  William  took  27  cents  from  his  purse,  and  there  were 
lU  left:  how  many  were  there  at  first? 

6.  The  sum  of  two  numbers  is  14,  and  their  difference  ig  2: 
what  are  the  numbers  ? 

Analysis. — ^The  diff*erence  of  two  numbers,  added  to  the 
lees,  will  gi^•c  the  greater.  Let  x  denote  the  less  number; 
then,  a;  4-  2,  will  denote  the  greater,  and  jc  +  aj  +  2, 
will  denote  their  sum,  which  is  14.  Then,  2a;  +  2  equals 
14;  and  2a;  equals  14  minus  2,  or  12:  hence,  x  equals 
12  divided  by  2,  or  6 :  hence,  the  numbers  are  G  and  8. 

VEIUFICATION. 

6  +  8  =  14,  their  sum;  and 
S  —  6  =     2,  their  difference. 

7.  Tlie  sum  of  two  numbers  is  18,  and  their  difference  6 : 
what  are  the  numbers  ? 

8.  James  and  John  have  26  marbles,  and  James  has  4  more 
than  John  :  how  many  has  each  ? 

9.  Jane  and  Lucy  have  16  books,  and  Lucy  has  8  more 
than  Jane  :  how  many  has  each  ? 

10.  William  bought  an  equal  number  of  oranges  and 
lemons ;  Charles  took  5  lemons,  after  which  William  had  but 
25  of  both  sorts :  how  many  did  he  buy  of  each  ? 

11.  Mary  has  an  equal  number  of  roses  on  each  of  two 
bushes ;  if  she  takes  4  from  one  bush,  there  will  remain  24 
en  both  :  how  many  on  each  at  first  ? 

i2.  The  sum  of  two  numbers  is  20,  and  their  difference 
b  6 :  what  are  the  numbers  ? 

Analysis. — ^If  x  denotes  the  greater  number,  a;  — -  6  will 
denote  the  less,  and  a;  +  a;  —  6  will  be  equal  to  20 ;  hence, 
2x  equals   20  +  6,   or  26,  and  x  equals  26  divided  by  2, 
eqiuiJF  13;  hence  the  numbers  are  13  and  7. 
2 


iJO  I  N  T  R  O  D  D  C  r  I  O  N  . 

WRITTEN. 

Let  X  denote  the  greater ;  then, 

aj  —  6  =  the  less ;  and 

a;  4-  a  —  6   =  20,  their  sum ;  hence, 

22;  =   20  +  6   =   26  ;  or, 

26 
X  =  —-  =   13  ;  and  13  —  0   ~   7. 

VERIFICATION. 

13  +  7   =   20  ;  and,  13  —  7   =  6. 

13.  The  sum  of  the  ages  of  a  father  and  son  is  60  ydflrs, 
and  their  difference  is  just  half  that  number  :  what  are  the'r 
ages? 

14.  The  sum  of  two  numbers  is  23,  and  the  larger  lacks 
1  of  being  7  times  the  smaller  :  what  are  the  numbers  ? 

15.  The  sum  of  two  numbers  is  50  ;  the  larger  is  equal  to 
10  times  the  less,  minus  5  :  what  are  the  numbers  ? 

16.  John  has  a  certain  number  of  oranges,  and  Charles 
has  four  times  as  many,  less  seven  ;  together  they  have  53  : 
how  many  has  each  ? 

17.  An  orchard  contains  a  certain  number  of  apple  trees, 
and  three  times  as  many  cherry  trees,  less  6  ;  the  whole  num- 
ber  is  30 :  how  many  of  each  sort  ? 


LESSON  IX. 

1.  If  a;  denotes  any  number,  and  1  be  added  to  it,  what 
will  denote  the  sum  ?  A?is.  ic  -f-  1. 

2.  If  2  be  added  to  jb,  what  will  denote  the  sum  ?     If  3 
be  added,  what  ?     If  4  be  added  ?  <fcc. 

If  to  John's  marbles,  one  marble  be  added,  twice  his  num- 
ber will  be  equal  to  1 0  :  how  many  had  he  ? 

Analysis. — Let  x  denote  the  number ;  then,  jc  +  1  will 
denote  the  number  after  1  is  added,  and  twice  this  number, 


MENTAL      !■:  X  E  R  C  I  ii  K  8  .  fi7 

or  2x  +  2,  will  be  equal  to  10.     If  2a;  +  2  is  equal  to  10, 
2x  will  be  equal  to  10  minus  2,  or  8 ;  or  x  vnH  be  equal  to  4. 

"WRFITEN. 

Lot  X  denote  the  number  of  John's  marbles;  then, 

a?  -f  1   =  the  number,  after  1  is  added ;  and 

2(a;  -f  1)   =   2a;  +  2   =   10  ;  licnce, 

8 
2a;  =   10  —  2  ;  or  a;  =   -   =4. 

VERIFICATION. 
2(4    +    1)    =    2    X    5    =    10. 

4.  Write  a;  +  2  multiplied  by  3.  Ans.  S{x  -f  2). 
Wljat  is  the  product  ?  Ans.  3a;-t-6. 

5.  Write  ar  -f  4  multiplied  by  5.  Ans.  b[x  -\-  4). 
What  is  the  product  ?  A7is.  bx  4-  20. 

6.  Write  a;  +  3  multiplied  by  4.  Ans.  A(x  +  3). 
What  is  the  product?  Ans.  4a;  -f-  12. 

7.  Lucy  has  a  certain  number  of  books ;  her  father  giv^a 
her  two  more,«svhcn  twice  her  number  is, equal  to  14  :  how 
many  has  she  ?  2^  (  x  ^  CI )       --    /  ^ 

8.  Jane  has  a  certain  number  of  roses  in  blossom  ;  two 
more  bloom,  and  then  3  times  the  number  is  equal  to  16  : 
how  many  were  in  blossom  at  first  ? 

9.  Jane  has  a  certain  number  of  handkerchiefs,  and  buys 

4  more,  when  5  times  her  number  is  equal  to  45 :  how  many 
had  she  at  first  ? 

10.  John  has  1  apple  more  than  Charles,  anl  3  times 
John's,  added  to  what  Charles  has,  make  15:  how  many 
ha»  each  ? 

Anai.tsi8. — Let  x  denote  Charles'  apples ;  then  x-\-\  will 
denote  John's ;  and  a;  -f  1  multiplied  by  3,  added  to  a^  or 
3ar  -f  3  H-  a;,  will  be  eqiial  to  15,  what  they  both  had;  hcnco, 
43  +  3  equals  15;  and  Ax  equals  15  minus  3,  or  12;  and 

05  =  4.     Write,  and  verify. 


28  INTRODUCTION. 

11.  James  has  two  marbles  more  than  William,  and  twice 
his  marbles  plus  twice  William's  are  equal  to  16  :  how  many 
has  each  ? 

12.  Divide  20  into  two  such  parts  that  one  part  shall  ex- 
ceed the  other  by  4. 

13.  A  fruit-basket  contains  apples,  pears,  and  peaches* 
tliere  are  2  more  pears  than  apples,  and  twice  as  many 
peaches  as  pear's;  there  are  22  in  all:  how  many  of  each 
tsort? 

14.  ^yiiat  is  the  sum  of  a;  +  3cc  +  2{x  +  1)  ? 

15.  What  is  the  sum  of  2(ic  +  I)  +  l{x  -{-  1)  +  x? 

16.  What  is  the  sum  of  cc  +  5(a;  +  8)  ? 

17.  The  sum  of  two  numbers  is  11,  and  the  second  is  equal 
to  twice  the  first  plus  2  :  what  are  the  numbers  ? 

18.  John  bought  3  apples,  3  lemons,  and  3  oranges,  for 
which  he  paid  27  cents;  he  paid  1  cent  more  for  a  lemon 
than  for  an  apple,  and  1  cent  more  for  an  orange  than  for  a 
lemon  :  what  did  he  pay  for  each  ? 

19.  Lucy,  Mary,  and  Ann,  have  15  cent*;  Mary  has  1 
more  than  Lucy,  and  Ann  t^vice  as  many  as  Mary  ? 


LESSO]^  X. 

1.  If  a,  denote  any  number,  and  1  be  subtracted  from  it, 
what  will  denote  the  difference?  Ans.  x  —  1, 

If  2  be  subtracted,  Avhat  will  denote  the  difference  ?  If 
3  be  subtracted  ?     4  ?  &c. 

2.  John  has  a  certain  number  of  marbles ;  if  1  be  taken 
away,  twice  the  remainder  will  be  equal  to  12;  how  many 
has  he? 

Analysis. — Let  x  denote  the  number  ;  then,  x  —  1  will 
denote  the  number  after  1  is  taken  away ;  and  twice  this 
number,  or  2 (a  —  1)   =  2x  —  2,  will  be  equal  to  12.     If  2aj 


MENTAL      EXBBOI8EB.  29 

diminislied  by  2  is  equal  to  12,  2a;  is  equal  to  12  plus  2,  or 
14  ;  hence,  x  equals  14  divided  by  2,  or  7. 

■WEITTKN. 

Let  X  denote  the  number ;  then, 

X  —  1   =  tlie  number  which  remained,  and 

2(a;  —  1)   =  2a5  -  2  =  12 ;  hence, 

14 
2a!  =  12  -f  2,  or  14 ;  and  a;  =  —  =  7. 

VEIUFICATION. 

2(7-1)  =  14-2  =  12;  also,  2(7  -  1)   =  2  X  6  =  12. 

3.  Write  3  times  aj  —  1.  Arts,  3(a;  —  1). 
What  is  the  product  equal  to?  Ans.  Zx  —  3. 

4.  Write  4  times  a;  —  2.  ^7? 5.  4(x  —  2). 
What  is  the  product  equal  to?  Ans.  Ax  —  8. 

5.  Write  5  times  x  —  5.  A7i8.  5(x  —  5). 
What  is  the  product  equal  to  ?  Ans.  5x  —  25. 

6.  If  x  denotes  a  certain  number,  will  a;  —  1  denote  a 
greater  or  less  number  ?  how  much  less  ? 

7.  If  as  —  1  is  equal  to  4,  what  will  x  be  equal  to  ? 

A71S.  4  +  1,  or  6. 

8.  If  aj  —  2  is  equal  to  6,  what  is  x  equal  to  ? 

9.  James  and  John  together  have  20  oranges ;  John  has 
6  le«8  than  James :  how  many  has  each  ? 

10.  A  grocer  sold  12  pounds  of  tea  and  coffee;  if  the  tea 
be  diminished  by  3  pounds,  and  the  remainder  multiplied  by 
2,  the  product  is  the  number  of  pounds  of  coffee :  how  many 
pounds  of  each? 

11.  Ann  has  a  certain  number  of  oranges ;  Jane  has  1  less 
and  twice  her  number  added  to  Aun'o  make  13 :  hew  many 
nas  each  ? 

Analysis. — Let  x  denote  the  number  of  oranges  which 
Ann  has;  then,  x —  1  will  denote  the  number  Jane  has, 


30"  INTRODUCTION. 

and  a;  4-  2a;  —  2,  or  3a;  —  2,  will  denote  the  number  both 
have,  which  is  13.  If  3a;  —  2  equals  13,  3a;  will  be  equal 
to  13  -+-  2,  or  15  ;  and  if  3a;  is  equal  to  15,  x  will  be  equal 
to  ]  5  divided  by  3,  which  is  5  :  hence,  Ann  has  5  oranges 
and  Jane  4. 

"WltlTTEN. 

Let    X   denote  the  number  Ann  has  ;  then, 

a;  —  1    =  the  number  Jane  has ;  and 

2  (a;  —  1)   r=  2a;—  2   =  twice  what  Jane  has;  also, 

a:  -H  2a;  —  2   =  3a;  —  2   =   13 ;  hence, 

15 
3a;  =    13  +  2   =   15:  or  a;  =  —  =  5. 

3 

VERIFICATION. 

5—4   =   1 ;  and  2  X  4  -f  5   =   13. 

12.  Charles  and  John  have  20  cents,  and  John  has  6  less 
than  Charles:  how  many  has  each  ? 

1 3.  James  has  twice  as  ma^ny  oranges  as  lemons  in  his  bas- 
ket, and  if  5  be  taken  from  the  whole  number,  19  will  re- 
main :  how  many  had  he  of  each  ? 

14.  A  basket  contains  apples,  peaches,  and  pears;  29  in 
all.  If  1  be  taken  from  the  number  of  apples,  the  remainder 
will  denote  the  number  of  peaches,  and  twice  that  remainder 
mAW  denote  the  number  of  pears :  how  many  are  there  of 
each  sort  ? 

15.  If  2a;  —  5  equals  15,  what  is  the  value  of  a;? 

16.  If  4a;  —  5  is  equal  to  11,  what  is  the  value  of  a;? 
IV    If  5a;  —  12  is  equal  to  18,  what  is  the  value  of  a;? 

18.  The  sum  of  two  numbers  is  32,  and  the  greater  ex- 
ceeds the  less  by  8  :  what  are  the  numbers  ? 

19.  The  sum  of  2  numbers  is  9 ;  if  the  greater  number 
bo  diminished  by  5,  and  the  remahidcr  multiplied  by  3,  the 
product  will  be  the  less  nmnber :  what  are  the  numbers? 

20.  There  are  three  numbers  such  tluit  1  taken  from  the 


URNTAL      BXEB0I6B8.  31 

first  will  give  the  second ;  the  second  multiplied  by  3  v^ill 
give  the  third ;  and  their  sum  is  equal  to  2G  :  what  are  the 
uunibei-8?     ' 

21.  John  and  Charles  together  have  just  31  oranges;  if 
1  be  taken  from  John's,  and  the  remainder  be  multiplied  by 
5,  the  }>roduct  will  be  equal  to  Charles'  number;  how  many 
has  each  ?    ''^  ^ 

22.  A  bosket  is  filled  with  apples,  lemons,  and  oranges,  in 
all  26 ;  the  number  of  lemons  exceed  the  apples  by  2,  and 
the  number  of  oranges  is  double  that  of  the  lemons :  how 
many  are  there  of  each  ?  5  " 


LESSON    XL 

1.  John  hiis  a  certain  number  of  apples,  the  half  of  which 
IS  e(iual  tn  10:  how  many  has  he? 

Analv.^is. — Let  x  denote  the  number  of  apples;  then, 
X  divided  by  2  is  equal  to  10;  if  one  half  of  x  is  equal  to 
10,  twice  one-half  of  a?,  or  a^  is  equal  to  twice  10,  which  is 
-•>;  hence,  x  is  equal  to  20. 

Note. — A  similar  analysis  is  applicable  to  any  one  of  thi* 
fractional  units.  Let  each  question  be  solved  according  to 
the  analysis. 

2.  John  has  a  certain  number  of  oranges,  and  one-tliird  of 
his  nuMiber  is  15:  how  many  h.as  he? 

3.  If  one-fifth  of  a  number  is  6,  what  is  the  number? 

4.  If  one-twelfth  of  a  number  is  9,  what  is  the  number?.^ 

5.  What  number  added  to  one-half  of  itself  will  give  a 
sum  equal  to  1 2  ? 

Analysis. — Denote  tin?  number  by  x;  then,  x  plus  one 
half  of  a;  equals  12.  But  jb  plus  one-half  of  a  equals  three 
halves  of  x:  hence,  three  halves  of  x  equal  12.  If  throe 
|,iKv>^  nf  X  equal  12,  one-half  of  x   equals  one-third  of  12, 


32 


INTRODUCTION. 


or  4.     If  one-half  of  x    equals  4,  x  equals  twice  4,  or  8, 
hence,  x  equals  8. 

WRITTEN. 

Let  X  denote  the  number;  then, 

1  3 

03  +  Tiic  =  ^^^  =  12  ;    then, 

-a  =     4,    or    JB  =  8. 


VERIFICATION. 

8-f-?rrr8   +   4    =    12. 

6.  What  number  added  to  one-third  of  itself  will  give  a 
sum  equal  to  12? 

7.  What  number  added  to  one-fourth  of  itself  will  give 
a  sum  equal  to  20  ? 

8.  What  number  added  to  a  fifth  of  itself  will  make  24  ? 

9.  What   number   diminislied   by  one-half  of  itself  will 
leave  4  ?     Why  ? 

10.  What  number  diminished  by  one-third  of  itself  wiD 
leave  6  ? 

11.  James  gave  one-seventh  of  his  marbles  to  William, 
and  tlien  has  24  left :  how  many  had  he  at  first  ? 

12.  What  number  added  to  two-thirds  of  itself  will  give 
a  siim  equal  to  20  ? 

13.  What  number  diminished  by  three-fourths  of  itself 
will  leave  9  ? 

14.  What  number  added  to   five-sevenths  of  itself  will 
make  24  ? 

15.  What  number  diminished  by  seven-eighths  of  itself 
w  ill  leave  4  ? 

16.  AVhat   number  added  to   eight-ninths  of  itself  will 
make  34? 


ELEMENTARY    ALGEBRA. 


CIIAPrER  I. 

DEFINITIONS   AND   EXPLANATORY  SIGNS. 

1.  Quantity  is  anything  which  can  be  increased, 
diminished,  and  measured;  as  number,  distance,  weight, 
time,  &c. 

To  measure  a  thing,  is  to  find  how  many  times  it  con- 
tains some  other  thing  of  the  same  kind,  Uiken  as  a  stand- 
ard.   The  assumed  standard  is  called  the  unit  of  measure. 

2.  Mathematics  is  the  science  which  treats  of  the 
measurement,  properties,  and  relations  of  quantities. 

In  pure  mathematics,  there  are  but  eight  kinds  of  quantity, 
and  consequently  but  eiglit  kinds  of  llNrrs,  viz.:  Units  of 
Number ;  Units  of  Currency ;  Units  of  Length ;  Units  of 
Surface;  Units  of  Volume ;  Units  of  Weight;  Units  of 
Time  ;  and  Units  oi  Angular  Measure. 

3.  Algebra  is  a  branch  of  Mathematics  in  which  the 
quantities  considered  are  represented  by  letters,  and  the 
operations  to  be  performed  are  indicated  by  signs. 

1.  What  19  quantity  ?  What  is  the  opcratioD  of  measuring  a  thing? 
Whit  is  the  assumed  standard  called  ? 

2.  What  is  Mathematics  ?  How  many  kinds  of  quantity  are  there  ic 
.he  pure  mathematics?    Name  the  unit«  of  thoe^  quantities. 

8.  What  is  Algebra? 
!♦ 


34  ELEMENTARY       ALGEBRA. 

4.  The  quantities  employed  in  Algebra  are  of  two  kinds, 
Known  and  Unknown  : 

Known  Quantities  are  those  whose  values  are  given ; 

and 
Unknown  Quantities  are  those  whose  values  are  rC' 
quired. 
Known  Quantities  are  generally  represented  by  the  lead 
ing  letters  of  the  alphabet,  as,  «,  ^,  c,  &c. 

Unknown  Quantities  are  generally  represented  by  tho 
final  letters  of  the  alphabet ;  as,  x^  y,  2,  &c. 

When  an  unknown  quantity  becomes  known,  it  is  often 
denoted  by  the  same  letter  with  one  or  more  accents ;  as, 
ik',  x'\  x".  These  symbols  are  read:  x  prime j  x  second; 
X  thirds  dbc. 

5.  The  Sign  of  Addiiton,  +,  is  called  plus.  When 
placed  between  two  quantities,  it  mdicates  that  the  second 
is  to  be  added  to  the  fii'st.  Thus,  a  -f  ^,  is  read,  a  plus  6, 
and  indicates  that  b  is  to  be  added  to  a.  If  no  sign  is 
wi-itten,  the  sign  -f  is  understood. 

The  sign  -f ,  is  sometimes  called  the  positive  sign,  and  the 
quantities  before  Avhich  it  is  written  are  called  j^ositive  quan- 
tities^ or  additive  quantities. 

6.  The  Sign  of  Subtraction,  — ,  is  called  mimis.  When 
placed  between  two  quantities,  it  indicates  that  the  second 
is  to  be  subtracted  from  the  first.     Thus,  the  expression, 


4.  How  many  kinds  of  quantities  are  employed  in  Algebra?  How  are 
they  distinguished  ?  What  are  known  quantities  ?  What  are  unknown 
quantities?  By  what  are  the  known  quantities  represented?  By  what 
are  the  unknown  quantities  represented  ?  When  an  unknown  quantity 
becomes  known,  how  is  it  often  denoted? 

5.  What  is  the  sign  of  addition  called?  When  placed  between  two 
quantities,  what  does  it  indicate  ? 

6.  What  is  the  sign  of  subtraction  called  ?  When  placed  between  two 
•■juautitifts,  what  does  it  indicate? 


DEFINITION       OK      TERMS.  35 

c  —  dy  road  c  minus  f7,  iiidicaltsthat  c?  is  to  be  subtracted 
from  c.  If  a  stands  for  6,  and  d  for  4,  then  a  —  d  \s  equal 
to  6  —  4 ,  which  is  equal  to  2. 

Tlie  sign  — ,  is  sometimes  called  the  negative  sign,  and  the 
qu:intiti*is  before  which  it  is  written  ai*e  called  negative  quaur 
FilieSy  or  aubtractive  quantities. 

T.  Tlie  Sign  qp  Multipucation,  x  ,  is  read,  mnltipUul 
bi/y  or  into.  When  placed  between  two  quantities,  it  indi- 
cates that  the  first  is  to  be  multiplied  by  the  second.  Thus, 
a  X  b  indicates  that  a  is  to  be  nmltiplied  by  b.  If  o  stand?? 
for  7,  and  b  for  5,  then,  a  X  b  \a  equal  to  7  x  5,  which  is 
equal  to  35. 

The  multiplication  of  quantities  is  also  indicated  by  simply 
writing  the  letters,  one  after  the  other ;  and  sometimes,  by 
placmg  a  point  between  them  ;  thus, 

a  X  b  signifies  the  same  thing  as  ab^  or  as  a.b. 

a  X  b  X  c  signifies  the  same  thmg  as  abc^  or  as  a.b.c. 

8.  A  Factou  is  any  one  of  the  multipliers  of  a  product. 
Factors  are  of  two  kinds,  numeral  and  literal.  Thus,  in  the 
expression,  5aic,  there  are  four  factors :  the  numeral  factor, 
5,  and  the  three  literal  fiictors,  a,  ^,  and  c. 

9,  The  Sign  of  Division,  -f-,  is  read,  divided  by.  When 
written  between  two  quantities,  it  indicates  that  the  first  is 
to  be  divided  by  the  second. 

7.  How  is  the  sign  of  multiplication  road  ?  "When  phiced  between  two 
quantities,  what  does  it  indicate?  In  how  many  ways  may  m;iltip)ieatiou 
be  indicated? 

8.  V«hat  is  a  factor?  IIow  many  kinds  of  factors  are  there T  Bow 
many  factors  arc  there  in  Zabc  ? 

9.  Uuw  is  the  sign  of  division  read?  When  wntten  between  two  quan- 
tiJicfl,  what  does  it  indicate?  IIow  many  ways  are  there  of  indicating 
division  ? 


86  ELEMENTARY       ALGEBRA. 

There  are  three  signs  used  to  denote  division.    Thus, 

a  -T-  5  denotes  that  a  is  to  be  divided  by  b, 

7  denotes  that  a  is  to  be  divided  by  h, 

a  I  b     denotes  that  a  is  to  be  divided  by  b. 

lO.  The  Sign  of  Equality,  =,  is  read,  equal  to.  Wlien 
M  rittcn  between  two  quantities,  it  indicates  that  they  are 
equal  to  each  other.  Thus,  tlie  expression,  a  +  6  =r  c,  in- 
dicates that  the  sum  of  a  and  ^  is  equal  to  c.  If  a  stands 
for  3,  and  b  for  5,  c  will  be  equal  to  8. 

ai.  The  Sign  of  Inequality,  >  <,  is  read,  greater 
thaii^  or  less  than.  When  placed  between  two  quantities, 
it  indicates  that  they  are  unequal,  the  greater  one  being 
placed  at  the  opening  of  the  sign.  Thus,  the  expression, 
a  >  ^,  indicates  that  a  is  greater  than  b\  and  the  expres- 
sion, c  <^  d^  indicates  that  c  is  less  than  d. 

12.  The  sign  .  * .  means,  therefore^  or  consequently. 

13.  A  Coefficient  is  a  number  written  before  a  quan. 
tity,  to  show  how  many  times  it  is  taken.     Thus, 

a-\-a-\-a-\-a-\-a  =  5a, 

In  which  5  is  the  coefficient  of  a. 

A  coefficient  may  be  denoted  either  by  a  number,  or  a 
letter.     Thus,  5x  indicates  that  x  is  taken  5  times,  and  ax 


10.  What  is  the  sign  of  equality  ?  When  placed  between  two  quanti- 
ties, what  does  it  indicate  ? 

n.  How  is  the  sign  of  inequality  read?    Which  quantity  is  placed  on  ^ 
the  side  of  the  opening  ? 

12.  What  does  .*.  indicate? 

18.  What  is  a  coefficient?  How  many  times  is  a  taKen  in  5a.  By 
what  may  a  coefficient  be  denoted  ?  If  no  coefficient  is  written,  what 
coefficient  is  understood  ?  In  5ar,  how  many  limes  is  ax  taken?  Ho-v 
niauy  tiraea  is  r  taken  ? 


DEFINITION      OF      TERMS.  37 

indicates  that  x  is  taken  a  times.  If  no  coefficient  is  ^\Tit- 
ten,  the  coefficient  1  is  understood.  Thus,  a  is  the  same 
as  la. 

14.  Ax  ExroxENT  is  a  number  wriiten  at  tlic  right  and 
Rl»ove  a  quantity,  to  indicate  how  many  times  it  is  taken  as 
a  factor.    Thus, 

a  X  a  is  written  a\ 


a  X  a  X  a 

(C 

o^ 

a  X  a  X  a  X  a 

(( 

a\ 

&0., 

Ac, 

m  which  2,  3,  and  4,  are  exponents.  The  expreshions  aro 
read,  a  square,  a  cube  or  a  third,  a  fourth ;  and  if  we  have 
o**,  in  which  a  enters  m  times  as  a  factor,  it  is  read,  a  to 
the  mth,  on  simply  a,  mth.  The  exj^onent  1  is  goneraUy 
omitted.  Thus,  a*  is  the  same  as  a,  each  denoting  that  a 
enters  but  once  as  a  factor. 

15.  A  Power  is  a  product  which  arises  from  the  multi- 
plication of  equal  factors.     Thus, 

a  X  a  =  a^   is  the  square,  or  second  power  of  a. 

a  X  a  X  a  =  a^  IS  the  cube,  or  third  power  of  a. 

axax«xa  =  a*  is  the  fourth  power  of  a. 

a  X  a  X  ...  .  =  a"  is  the  mth  power  of  a. 

16.  A  Root  of  a  quantity  is  one  of  the  equal  factors, 
riie  radical  sign,  \/  ,  when  placed  over  a  quantity^  indi- 
cates that  a  root  of  that  quantity  is  to  be  extracted.  Tlie 
root  is  indicated  by  a  number  written  over  the  radical  sign, 

14.  What  is  an  exponent?  In  a',  how  many  times  is  a  taken  as  a  fao- 
or?     When  no  exponent  is  written,  what  is  understood? 

16.  What  is  a  power  of  a  quantity?  What  is  the  third  power  of  2» 
Of  4  ?    Of  6  ? 

16.  What  is  the  root  of  a  quantity?  What  indicates  a  root?  What 
indicates  the  kind  of  root?  Wh.it  is  the  index  of  tho  pquare  root?  0( 
the  ctibc  root  ?     Of  the  mth  root  ? 


38  ELEMENTARY       ALGEBRA. 

called  an  index.     When  the  mdex  is  2,  it  is  generally  omit- 
ted.    Tims, 

\/a^  or  y/a,  indicates  tlie  square  root  of  a. 

ya  indicates  tbe  cuhe  root  of  a. 

^/a  indicates  the  fourth  root  of  a. 

\/a  mdicates  the  fnih  root  of  a. 

]  ?  .     An  Atx^.ebraic  Expression  is  a  quantity  written  in 
algebraic  language.     Thus, 

„    j  is  the  algebraic  expression  of  ttree  tunes 

(      tbe  number  denoted  by  a  ; 

-   2  ^  '^s  the  algebraic  expression  of  five  times 

I      the  square  of  a  ; 

is  the  algebraic  expression  of  seven  times 

la^b^  -j      the  the  cube  of  a   multiplied  by  the 

square  of  b ; 

is  the  algebraic  expression  of  the  differ- 

Sa  —  5h -l      ence  between  three  times  a  and  five 

times  b ; 

r  is  the  algebraic  expression  of  twice  the 

„  „       „    ,        ,,„         square  of  a.  diminished  by  three  times 
2a^  —  Sab  +  4h^  ^      ^/  i     1    i-      ^      j  .   i  ^ 

J       the  product  oi  a  by  t>,  augmented  by 

I       four  times  the  square  of  b. 

18      A  Term  is  an  algebraic  expression  of  a  sm gle  quan- 
tity.    Thus,  3a,  2ab,   —  5cfb\  are  terms. 

19.     The  Degree  of  a  term  is  the  nmnber  of  its  literal 
factors.     Thus, 

„    j  is  a  term  of  the  first  degree,  because  it  oontains  but 
(      one  literal  factor. 

17.  What  is  an  algebraic  expression 

1 8.  What  is  a  term  ? 

1 9.  What  is  tlio  dogree  of  a  term  ?  What  detenniriop  the  (Vgrpe  of  a  term  ? 


Trf'ij 


DEFINITION      OP      TERMS.  39 

•  j  is  of  tlie  second  degree,  because  it  contains  two  lite- 
(      ral  factors, 
is  of  the  fourth  degree,  because  it  contains  four  literal 
factors.     Tlie  degree  of  a  term  is  determined  by 
the  sum  of  the  exponents  of  all  its  letters. 

20.  A  MoKOiHAL  is  a  single  term,  unconnected  with  any 
other  by  the  signs  +  or  —  ;  tlms,  3a^  Slt^a,  are  monomials. 

a  I.  A  PoLYNOiTiAL  is  a  collection  of  terms  connected 
()}•  the  signs  -H  or  —  ;  as, 

3a  —  5,  or,  2a^  —  db  +  4h^. 

22.  A  Binomial  is  a  poljTiomial  of  two  temis ;  as, 

a  -f  6,  3a2  _  c^  6ab  —  c^. 

23.  A  Trinomial  is  a  polynomial  of  three  terms  ;  as, 

abc  —  a^  +  c\  ab  —  (jh  —  f. 

21.  Homogeneous  Terms  are  those  which  contain  the 
p;»iiK' number  of  literal  factors.  Thus,  the  terms,  abc^  —  «\ 
4-  c^,  are  homogeneous ;  as  are  the  terms,  ab^  —  gh. 

25.  A  Polynomial  is  tiomogeneous,  when  all  its  terms 
!''  homogeneous.    Tlius,  the  polpiomial,  abc  —  a^  +  c^,  is 

iJiogeneous;  but  the  polynomial,  ab  —  gh  —  f  is  not  ho- 
iii<»geneou8. 

26.  Snrn.AR  Terms  are  those  which  contain  the  same 
Ht.'ial  factors  affected  \^'iih  the  same  exponents.     Thus, 

*lah  4-  3rt/>  —  2a^ 

- 1».  What  19  a  monomial  ? 

21.  What  is  a  polynomial? 

'Zl.  What  is  a  binomial? 

.:;.  What  is  a  trinomial? 

.  \.  What  are  homojjpneous  terms  ? 

•J.").  When  i«  a  polTnomial  homogeneous  ? 

L'-i.  What  nrp  «:niil:'.r  tpnn'J  ? 


40  ELEMENTAKY      A  L  O  E  B  E  A. . 

are  similar  terms ;  and  so  also  are, 

but  the  terms  of  the  first  polynomial  and  of  the  last,  are  not 
similar. 

'   27.     The  Yinculttm,  ,   the  Bar   \  ,   the  Parens 

thesis^  (  ) ,  and  the  Brackets^  [  ] ,  are  each  used  to  con- 
nect several  quantities,  which  are  to  be  operated  upon  in  the 
Fame  manner.    Thus,  each  of  the  expressions, 

a    X 
'a^i^^c  X  Jc,         f  5  (a  +  ^>  +  c)  X  aj, 

and  [a  +  5  4-  c]  x  j», 

indicates,  that  the  sum  of  a,  J,  and  c,  is  to  be  multiplied 
by  JB. 

28.     The  RECirROCAL  of  a  quantity  is  1,  divided  by  that 
quantity;  thus. 


are  the  reciprocals  of 

.        7  ^ 

a,    a  -{•  0 .    -• 
c 

29.  The  Numerical  Yalue  of  an  algebraic  expression, 
is  the  result  obtained  by  assigning  a  numerical  value  to  each 
letter,  and  then  performing  the  operations  mdicated.  Thus, 
the  numerical  value  of  the  expression, 

ah  -\-  he  +  cL 
when,     a  =    i,    J  —  2,    c  =  3,    and    d  =:  4,    is 

1x2  +  2x3  +  4  =   12; 
by  performing  the  indicated  operations. 


27.  For  what  is  the  vincular  used?    Point  out  the  other  ways  in  which 
this  may  be  done  ? 

28.  What  is  the  reciprocal  of  a  quantity? 

29.  What  is  the  numerical  value  cf  an  algebraical  exprepsion?  1 


ALOBBRAIC      EXPRESSIONS  4l 

EXAMPLES  IX   WETTING  ALGEBSAIC  EXPEESSIONS. 

1.  Write  a  added  to  b.  Ans.  a  -f  ft« 

2.  Write  b  subtracted  from  a.  Ans.  a  —  b. 

Write  the  following : 
8.  Six  times  the  square  of  a,  minus  twice  the  square  of  b. 
4.  Six  times  a  multiplied  by  b,  diminished  by  5  times  c 
cube  multiplied  by  d. 

6.  Nine  times  a,  multiplied  by  c  plus  d,  diminished  by 

8  times  b  multiplied  by  d  cube. 

6.  Five  times  a  minus  b,  plus  6  times  a  cube  into  b 
cube. 

I.  Eight  times  a  cube  into  d  fourth,  into  c  fourth,  plus 

9  times  c  cube  into  d  fifth,  minus  C  times  a  into  b^  into  c 
square. 

8.  Fourteen  times  a  plus  J,  multiplied  by  a  minus  5, 
plus  6  times  a,  into  c  plus  f/. 

9.  Six  times  a,  into  c  plus  t7,  minus  5  times  ft,  into  a  plus 
c,  minus  4  times  a  cube  ft  square. 

J     1 0.  Write  a,  multiplied  by  c  plus  d,  plus  /  minus  ^. 

II.  Write  a  divided  by  ft  +  c.    Three  ways. 

12.  Write  a  —  ft  divided  by  a  +  ft. 

13.  Write  a  polynomial  of  three  tenns;  of  four  terms;  of 
five,  of  sb:.  ^ 

14.  Write  a  homogeneous  binomial  of  the  first  degree;  of 
the  second ;  of  the  tliird ;  4th  ;  6th  ;  6th. 

15.  Write  a  homogeneous  trinomial  of  the  first  degree; 
with  its  second  and  third  terms  negative;  of  the  second 
degree;  of  the  3rd;  of  the  4th. 

16.  Write  in  the  same  column,  on  the  slate,  or  black-board, 
a  monomial,  a  binomial,  a  trinomial,  a  polj-nomial  of  four 
terms,  of  five  terms,  of  six  tenns  and  of  seven  tenns,  and  aJJ 
of  the  sanie  degree. 


42  ELEMENTARY       ALGEBRA. 


INTKEPRETATION    OF   ALGEBRAIC   LANGUAGE. 

u 

Find  the  numerial  values  of  the  folloTving  expressions, 
when, 

a  =  1,    ^  —  2,    c  =  3,    c?  =.  4. 

1.  ab  +  he.  A?is,  8.  ' 

2.  a  +  be  -{-  d.  A?is.  11. 

3.  ad  -\-  b  —  c.  Ans.  '6. 

4.  ab  •\-  be  —  d.  Ans.  4. 

5.  (a  +  b)  e^^-  d.                           ,    •  -  Ans.  23. 

6.  (a  +  b)  {d  —  h.)                                              ^  Ans.  6. 

7.  (a^.+  «c7,)  c  -\-  d.  Ans.  22. 

8.  {ab  -h  o)  (ac?  —  a).  Ans.  15. 

9.  ^a2^2  _\2(a  +  c?  -1-  1).  ..4/i6\  0. 

10.  ^  ^  X  («  4-  <^)  ^;z5.   10. 

tt^":}.    52   _|_    c2  a3  _^    J3  -|_    c^  -   f/  . 

11.  r X r •  Ans.  32. 

7  2 

a54  _  c  _  ^3       J^2  -  h  -\-  d^ 

12. x^= -r-^- — ^^  A?is.     4. 

6  33 

Fmd  the  numerical  values  of  the  folio  ^^-ulg  expressions, 
when, 

a  =  4,    b  =  Sy    c  =  2,    and  d  =  1.. 

13.  ^   -   7:  +  c  —  d.  Ans.     2. 

^ ,    ^/''d)       a  —  d\ 

U.  5f  ^ 3~/'  ^^15.  15. 

15.  [(rt2Z>  +  l)cn    -f-   (a^b  +  cZ).  ^715.     1. 

16.  4(abc  -  -^l    X    (30c3  —  aPd^).  A?is.  11088. 
,^     «  +  ft  +  c     ,    abed   ,    4^2+  ft^  -  <?^ 

^^'  ^Try+^  +  -^  +  — 5^TT-'     ^'^^-  ''^ 

,,.  IM^^^  „  --^  ^^'  X  a^^W3      ^...  3405, 
Sc^  ^       2         Lahd 


A  DDI  nON. 


43 


CIIAFfER  n. 

FUNDAMENTAL     OPEfiAlIONg. 
ADDITION. 

30.  Addition  is  the  operation  of  finding  the  simplest 
otjuivalent  expression  for  the  abnegate  of  two  or  more 
algebraic  quantities.     Such  exj)ression  is  called  their  Sum. 


When  t/ie  terms  are  similar  and  have  like  sif/ns. 

4 
31.     1.  Wliat  is  the  sum  of  a,  2a,  3a,  and  4a? 

Take  the  sum  of  the  coefficients,  and  annex  the 
literal  parts.  The  first  tenn,  a,  has  a  coefficient, 
1,  onderstood  (Art.  13). 


a 

4-  2rt 
-f  3a 
4-    4a 


2.  Wliat  is  the  sum  of  2aft,  3ai,  6aJ,  and  ab: 

When  no  sign  is  writttcn,  the  sign   +   is  under- 
stood (Art  5). 


4-  10a 

2al> 

Sab 

Qab 

ab 


Add  the  follownng : 

(3.)  (4.) 

a  Sab 

a  lab 


\2ab 


4-  2a 


\5ab 


(5.) 
lac 
5ac 

I2ac 


(6.) 

-f  ^abc 
Sabc 

+  labc  • 


..u.  What  ia  addition  ? 

31.  What  is  the  rule  for  addition  when  the  terms  are  Biniilarand  hftve 


M  ELEMENTARY      ALGEBRA. 

.0')  (8.)  (9.)  (10.) 

—  Zabq  —  Sad  —  2adf  —    9abd 

—  2abc  —  2ad  —  Qadf  —  I5abd 

—  5abc  —  had  —  '^adf  —  2\abd 
Hence,  when  the  terms  are  similar  and  have  like  signs : 

RULE. 

Add  the  coefficients^  and  to  their  sum  prefix  the  common 
sign  y  to  this,  annex  tJite  common  literal  part, 

EXAMPLES. 

(11.)                           (12.)  (13.) 

9ah  +    ax  8ac^  —  db^  \ba¥c^  —  I2abc^ 

Sab  +  Sax  lac^  —  8b^  12ab^c*  —  15abc^ 

\2ab  +  4aa;  Sac"^  —  db'^  ab^c*  —      abc^ 

When  the  terms  are  similar  and  have  unlike  signs, 

32.  The  signs,  +  and  — ,  stand  in  direct  opposition  to 
each  other. 

If  a  merchant  writes  +  before  his  gains  and  —  before  ?iis 
losses,  at  the  end  of  the  year  the  sum  of  the  plus  numbers 
will  denote  the  gains,  and  the  sum  of  the  minus  numbers 
the  losses.  If  the  gains  exceed  the  losses,  the  difference, 
which  is  called  the  algebraic  sum,,  will  be  plus ;  but  if  the 
losses  exceed  the  gains,  the  algebraic  sum  will  be  minus. 

1.  A  merchant  in  trade  gained  $1500  in  the  first  quarter 
of  tlie  year,  $3000  in  the  second  quarter,  but  lost  $3000  in 
the  third  quarter,  and  $800  in  the  fourth  :  what  was  the  re- 
sult of  the  year's  business? 

1st  quarter,     +1500  3d  quarter,     —3000 

2d         "  3000  4th       "  —     800 

-h  4500  —  3800 

+  4500  —  3800   =    +  YOO,  or  $700  gain. 

32.  What  is  the  rule  when  the  terms  are  s'milar  and  havt;  unlike  sigiia  ? 


ADDITION.  45 

2.  A  merchant  in  trade  gained  llOOO  in  the  first  quarter, 
and  $2000  the  second  (jiiarter ;  in  tlie  tliird  quarter  he  lost 
llaOO,  and  in  the  fourth  quarter  $1800.-:  what  was  the  result 
of  the  year's  business  ? 

iPt  quarter,     +  1000  3d  quarter     —  1500 

2d             *'       -t-  2000  4th       "           --  1800 

H-  3000  —  3300 

+  3000  —  3300  =  —  300,  or  $300  loss. 

3.  A  merchant  in  the  first  half-year  gained  a  dollars  and 
lost  b  dollars ;  in  the  second  half-year  he  lost  a  dollais  and 
gained  b  dollars :  what  is  the  result  of  the  year's  business  ? 

Ist  half-year,  -\-  a  —  6 

2d        "  -  a  +  b- 

Result,  0  0 

Hence,  the  algebraic  sum  of  a  positive  and  negative  quan- 
tity  is  their  arithmetical  diffcre7ice^  with  the  sign  of  tlie 
greater  jyrefixed.     Add  the  following : 

^ah  Aach"^  —  Aa'^b'^c^ 

dab  —  Sacb^  +  6aV)^c^ 

-  Gab  aclP-  —  la^h^c^ 

bab  —  dacb'^  0 

Hence,  when  the  terms  are  similar  and  have  unlike  signs : 

I.    Write  the  similar  terms  in  the  same  column  : 
n.  Add  the  coefficients  of  the  additive  terms ^  and  also 

the  coefficients  of  the  subtractive  terms  : 
UL  Take  the  difference  of  these  sums,  prefix  the  sign 

of  the  greater,  and  then  annex  the  literal  part, 

EXAMPLES. 

1.  What  is  the  sum  of 

2a2^>3  _  5^2^3  4-702^3  ^  oa^ft^  -  Wa^l^'t 


46 


KLKMENTAliY       ALGEBRA 


Having  written  the  similar  teinns  in  the  same 
cohimn,  we  find  the  sum  of  the  positive  coeffi- 
cients to  be  15,  and  the  sum  of  the  negative 
coefficients  to  be  —  16  :  their  difference  is  ■—  1 ; 
hence,  the  sum  is  —  a^h^. 


—  5r/2*3 

-  am. 


2.  "Wliat  is  the  sum  of 
'a/h  +  ^a^h  —  ^a^b  +  4«^^>  -  Qo^b  —  a^bt 

3.  What  is  the  sum  of 

1 1a?bc^  —  4a^bc^  H-  Qa^bc''  -  8a^c^  -f  1 1  a^bc^  ?    Ans.  I  la^bc", 

4.  What  is  the  sum  of 

4a2j  _  8^2^  -  ^a'b  -f  11«2^,? 

5.  What  is  the  sum  of 
labc^  —  abc^  —  nabc?  +  ^ahc^  -f  Q>abcn 

6.  What  is  the  sum  of 
9c^3-  5W>3—  8ac2-h  20c63  4-  Gac^  -  24c^1»       Ans,  X  og 


Ans,  1d?-h, 


Ans,  —  2«*V> 
Ans.  \'^abc\ 


To  add  any  Algebraic  Quantities, 

33.     1.  What  is  the  sum  of  3a,  55,    and    —  2c? 
Write  the  quantities,  thus, 

3a  +  55  —  2c; 

which  denotes  their  sum,  as  there  are  no  similar  terms, 

2.  Let  it  be  required  to  find  the  sum  of  the  quantities, 

3a2  __  8a j  -\-  /h^ 

lab  —  552 

ba^  —  bah,  —  46^ 


38.  What  is  tlie  rule  for  the  addition  of  any  lilgc'brain  quantities? 


ADDITION.  47 

From  the  preceding  examples,  we  have,  for  tlie  addition 
of  algebraic  quantities,  the  following 

RULE. 

L    Write  the  quantities  to  be  added,  placing  similar  terms 
ill  the  same  colum?i,  and  giving  to  each  its  proper  sign: 

TL   Add  up  each  column  separately  and  then  annex  tht 
dissimilar  terms  with  their  proper  signs, 

EXAMPLES. 

1.  Add  together  the  polynomials, 
3a?  —  2/>«  -  Aab,  5a^  -  b^  +  2ab,  and  dab  -  3c^  -  2^. 


Tlie  terra  Sa^  being  similar  to 
5a\  we  yn-iie  8a^  for  the  result 
of  the  reduction  of  these  two 
terms,  at  the  sjime  time  slightly 
crossing  them,  as  in  the  first  terra. 


5/r-  +  lib  -    V- 

4-  d^(h  —  ^^  —  3^ 
8a2  -I-    ab  -  bb^  -  Z(^ 


Passing  then  to  the  term  —  Aab,  which  is  similar  to 
4-  2ab  and  -h  dab,  the  three  reduce  to  +  ab,  which  if. 
phu'cd  after  8a?,  and  the  tenns  crossed  like  the  first  terra. 
Passing  then  to  the  terms  involving  6?,  we  find  their  euni 
to  be  —  bh^,  after  which  we  write  —  3c?. 

The  marks  are  dra\vn  across  the  terms,  that  none  of  them 
may  be  overlooked  and  omitted. 

(2.)  (3.>  (4.) 

labc  +  Ocrz  8aa;  -\   db  12o  —    6c 

—  dabc  —  3aa;  bax  —  96  —  3a-  —    9c 

Aakc  +  6aa;  13aa;  —  6A.  9a  —  16& 

Note. — If  a  =  5,  6  =  4,  c  =  2,  a;  =  1,  what  are  the 
numerical  values  of  the  several  sums  above  found  ? 


48  ELEMENTARY       ALGEBSA. 

(5.)  (6.)  (7.) 

9«  -f  /  Gax  —    Sac  Saf  +    (7    -f  «i 

—  Qa  +  (/  —  lax  —    Qac  ag  —  ^af  —  m 

—  2a  —  f  ax  4-  1 7ac  ah  —    ag  -\-  Zg 

-(8.)  (9.)         ' 

—  3a:  —  3a^  —  5c  —  7cc2  —  l^aca;  +  Wd'h'^G^ 

hx  —  9ab  —  9e  —  4x^  +   kacx  —  20a^b^c^ 


^  (10.)  ^'^  (11.) 


=7 — =i:t-t^  -i- 


22h  —  3c  — ,  7/  +  Sg  19ah^  +  3a'S^  —  Sax^ 

—  3A  -f  8c  —  2/  —  9^  -f  5a;       -  llah^  -  9a^b*  +  9ag^^ 

(12.)^^  (13.) 

7a;  —  9?/  +  52  +  3  —    <7  8a  +    ^ 

—  a;  —  3y  —  8  —    ^  2a  —    6  +    c 

—  a;+y  —  3s  +  l  +  7^  —  3a +5  +2c^ 

—  2a;  4-  63/  +  32  —  1  —    ^  —  65  —  3c  +  3<^ 

U.  Add'ltogether  —  5  -|-  3c  —  c7  —  115c  +  6/  —  5<7,  35 

—  2c  —  3d  —  e  ^  27/,    5c  —  86?  +  3/  —  1g,    —  7^—  6c 
-f  l7(?-f  96  - -cN/-i-  11^,    —  35  —  5d—2e-{-6f—9g-\-h. 

Ans.  —  85  —  109e  +  37/  -  10^  +  A. 

15.  Add  together  the  polynomials     7a25  —  3c?5c  —  852<3 

—  9c3  +  cd\    Sabc—  ^a?b  +  Sc^  —  452c  +  cd\  and  4a25 

—  8c3  4-  952c  —  36?^ 

Ans.  6a^b  4-  5a5c  —  35^^  —  14c3  +  2cd^  —  3d^. 

16.  What  IS  the  sum  of,    ^a^bc  +  65a;  —  4a/,    —  3a25c 
-05^7  +  14a/,    -  a/+ 95a;  +  2a25c,    +  6a/— 85a;+ 6a25c? 

Ans.  lOa'^bc  +  5a;  +  15a/. 

17.  "What  is  the  sum  of    a^n^  +  Za^m  +  5,     —  Qa'^11^ 
—  6a^m    -  5,    +  95  —  9a^m  —  ba^n?  ? 

Ans.   —  lOahi^  —  ^2ahn  +  95. 


(22.) 

(23.) 

Hn  -j-  b) 

5{a-  -  c^ 

3(a  -1-  />) 

-  4(a2  -  c^) 

2{a.  -f-  b) 

-  1(^2  _  c2) 

ADDITION.  ^ 

18.  Wliat  is  the  Bum  of  ia^b'^c  ~  \6a*x  —  9<ix^ds 
-r  Qa^b^c  —  Gaa^d  4-  lla'Xy    -h  lOoic^r^  —  </*«  —  9a^b'^c? 

Ans.  a^b^c  +  ax^cl 

19.  Wliat  is  the  sum  of    -  7^  +  35  -f  4/7  -  26     -h  3/7 

—  36   f  26?  yl7is-  0. 

20.  TTliat  is  the  sum  of,    ab  +  807/  —  m  —  7?,     —  fia*// 

—  Ow   f  \\n  +  cf7,    4-  Sxy  +  47/i  —  10?^  +//7? 

21.  What  is  the  sum  of  ixtj  +  n  +  Qax  -f  9am,  —  (Ja;y 
4-  6n  —  6cra;  —  8rtm,    2a;y  —  1?i  -{-ax  —  «mV     Ans.   4-  w«. 

(24.) 
9(c3  -  aP) 

-  10(r^  -  g/O 

T(a  4-  b)  ^  6(o3  -  af^} 

Note.  Tlie  quantity  within  tlic  parenthesis  must  be 
regarded  as  a  single  quantity. 

25.  Add  3«(//2  —  /i2)  —  2a(^2  —  //2)  4-  4a(./72  _  A?) 
4-  8a((7«  —  A^)  -  2a(/72  -  A^).  Ann.  \\a(ff  —  bT). 

26.  Add  Zc{n^c  -  b^)  -  Oc(aV  -  6^)  -  Vc'lrt^c  -  6') 
4-  15c(a2c  —  6^)  4-  c(flr2c  _  6").  Am.  Sc{a^c  -  //'). 

34.  In  algebra,  the  term  add  docs  not  always,  as  in 
aritlimetic,  convey  the  idea  of  augmentation ;  nor  the  temi 
sum,  the  idea  of  a  number  numerically  greater  than  any  of 
the  numbers  added.  For,  if  to  a  we  add  —  6,  we  have, 
a  —  bt  which  u?,  arithmetically  speaking,  a  diiference  be- 
tween the  number  of  units  expressed  by  a,  and  the  number 

S4.  Do  the  words  add  and  *M7n,  in  Algebra,  convey  the  same  ideas  aa 
L\  Arithmetic.     What  is  the  algebraic  sum  of  9  and   —  4  ?     Of  8  oiid 

-  2  ?     May  an  algebraic  sum  be  negative  ?     What  is  the  sum  of  6  and 

-  10?     IIow  an^  puch  sums  dirtingnbhod  from  arithntciiral  punisf 

3 


50  ELEMENTARY       ALGEBRA. 

of  units  expressed  by  h.  Consequently,  this  result  is  un 
merically  less  than  a.  To  distinguish  this  sum  from  an 
arithmetical  sum,  it  is  called  the  algebraic  sum. 


SUBTRACTION. 

35.  SiTBTKACTioN  is  the  operation  of  finding  the  differ- 
ence between  two  algebraic  quantities. 

36.  The  quantity  to  be  subtracted  is  called  the  Subtror 
hend  ;  and  the  quantity  from  which  it  is  taken^  is  called  the 
Minuend. 

The  dljference  of  two  quantities,  is  such  a  quantity  aa 
added  to  the  subtrahend  will  give  a  sum  equal  to  the  min- 
uend. 

EXAMPLES. 

1.  From  I7a  take  6a. 

1    OPEnATION. 

In  this  example,  l7a  is  the  mmuend,  and  6«  j^^ 

the  subtrahend:  the  difference  is  11a;  because^  6(j 

11a,  added  to  6a,  gives  17a.  I         j^ 

Tlie  difference  may  be  expressed  by  writing  the  quantities 

thus: 

17a  —  ^a  z:^  11a; 

in  which  the  sign  of  the  subtrahend  is  changed  from  4 
to  — . 


2.  From  \hx  take  —  9a;. 

The  dljference^  or  remainder,  is  such  a  quantity, 
as  being  added  to  the  subtrahend,  —  9a;,  will 
give  the  minuend,  15a;.  That  quantity  is  24a;, 
and  may  be  found  by  simply  changing  tJie  sig7i 


CFEEATIOH 

15.a; 
240 


OPEBATIOH. 

4-  q  ~  5 
Rem.  lOax  —  a  +  6 
add  +  a  —  b  X 

lOax  > 


SUBTRACTION.  51 

of  the  subtrahend,  and  adding.     Whence,  we  may  write, 
15a;  —  {_  9a;)   =  24a;. 

3.  From  lOoa;  take  a  —  b. 

The  difference^  or  remainder^  is  such  a  quantity,  as  added 
to  a  —  ^,  will  give  the  minuend,  lOaa;:  what  is  tliat  qaair 
tity? 

If  you  change  the  signs  of  both 
ttnns  of  the  subtrahend,  and  add, 
you  liavc,  lOax  —  a  -h  b.  Is  this 
the  true  remainder  ?  Certainly. 
For,  if  you  add  the  remainder  to 
the  subtrahend,  a  —  b^  you  obtain 
the  minuend,  lOax. 

It  is  plain,  that  if  you  change  the  signs  of  all  the  terms 
of  the  subtrahend,  and  then  add  them  to  the  mmuend,  and 
to  this  result  add  the  given  subtrahend,  the  last  sura  can  be 
no  other  than  the  given  minuend ;  hence,  the  Jirst  result  is 
the  true  difference,  or  remainder  (Art.  36). 

Uence,  for  the  subtraction  of  algebraic  quantities,  we  have 
the  following 

BULE. 

L  Write  t/te  terms  of  the  subtrahend  under  those  of  the 
miyiuend^  placiyig  similar  terms  in  the  same  column  : 

IL  Coficeive  the  sigyis  of  all  the  terms  of  the  subtrahend 
to  be  clhanged  from  ■\-  to  —^  or  from  —  ^o  -j-,  and  t/veti 
proceed  as  in  Audition, 


EXAMPLES 

OF 

M0X0MIAIJ8. 

(1.) 

(2.) 

(3.) 

From 

Sab. 

6aaj 

9abo 

take 

2ab 

Sax 

lahc. 

Rem. 

ab 

Sax 

2«//x? 

62 


ELEMENTARY       ALGEBRA. 


From 
taks 
Rem. 


I<Vom 
take 
Rem. 


(4.) 

Sax 

8c 


(5.) 

SaWc 
l4a'Pc 

(8.) 
4abx 
9ac 


(0.) 

1  laWx 
(9.) 

ax 


Sax  —  8c        4abx  —  9«c        2  am 


10.  From 

11.  From 

12.  From 

13.  From 

14.  From 

15.  From 

16.  Frora 

17.  From 
13.  From 

19.  From 

20.  From 

21.  From 

22.  From 

23.  From 

24.  From 

25.  From 

26.  From 

27.  From 

28.  From 

29.  From 


9^262  take^  Sa^b"^. 
IGa^xy  take  —  Iba^xy. 
\2a'if  take  8ay. 
l^a^x^y  t^k.Q  —  \%a^x^y. 
Vya^b^  take  3a^b^. 
la^b^  take  Qa^b"^, 
Sab^  take  aW. 
x'^y  take  y'^x. 
dx'^y^  take  a*?/. 
Sa^y^x  take  ccys. 
ga'-^^^  take  —  Sa:^b'^. 
14ay  t^e  —  20rt"-^y^ 

—  24a^5^  take  16aV)^- 

—  13cc2y  take   ^  14»;2y^ 

—  Ala^xhj  take  —ba^x-y. 

—  94a2a;2  take  Sa^ajZ. 


-4/zs. 


-4?75^  6a^J^ 

A?is.    4a^y^. 
Alls,   dla^x^y, 

3«2J3   _    3^3§2^ 


-4^15.  7a^/>' 


a  +  JB^  take 
a^  +  i^  take 


y3 


-  6a'b^. 
Ans.   Sab"^  —  a^b\ 
Ans.  x'^y  —  y'^x. 
Ans.    Sx'^y^  —  xy. 
Ans.   SaH/'^x  —  xyz. 
Ans.    Ua'^b^. 
Ans.    34a^y^. 
Ans.    —  40a^b^. 
Ans.   x^y*i 
Ans.    —  42(Px^y. 
Ans.    —  97a^x^ 
Ans.   a  -I-  a;2  4-  y''-. 


a?  —  b^.        Ans    2a^  +  2b^ 


IQa^x^y  take  ~  Ida'^x^y.    Ans.   +  Sa'^x^y. 


take  a^  4-  a^^. 


A?is.    —  2x^. 


SUBTRACTION.  53 


OBNEBAL      EXAMPLES. 

(1.)  flj  (!•)  n 

FromQac  —  5ab+    c*  *'s^         6ac  —  5tf/>  +    c* 

take  (3^c  ->?  3tf^  4::  7c^  ^        sM     —  ^>rrc  —  3ab  —  1c 

Rem.  3ac  —  8a^  +    c^  —  Vc.  ^^  _^ 

(2.) 

From     Gax  —  a  +  3b^ 
take        9aa;  ^x  -V  b^ 


Rem.  —  3aa?  —  a  +  x  +  2b\ 

(4.) 
From     5a3  — 4a=i+    2b^c 
take   -f  2a3if3a2^^    86^c 


-    • ' 

10  -  8a5  + 
(3.) 

c2- 

y<3 

Qf/x 

-  3a:2  +  55 

-  2/a; 

-i-3      -ha 

5?/a; 

—  na:2  4-  3  -t-  66  - 

-a. 

(5.) 

4( 

lb  —    cd+. 

;a2 

5a^>  -  4<Y?  4- ; 

Ja2_|. 

56» 

Rem.      1a^  —  la^b  +  1 1  b'^c.  -    ab -{-  :'>.  -d  -  5^2. 

6.  From  a  +  8  take  c  —  5.  -47i5.  a  —  c  +  13 

7.  From  Ca^  —  ]5  take  Oa^  v^  30.     ^7i«.   —  da^  -  45 

8.  From  Gary  —  Sa^^a  take  -4-  Ixy  -f  «V. 

-^1«5.    I3xy  —  lifc^ 

9.  From  a  -\-  c  take  —  a  —  c.  Ansl  2a  -\-  2c 

10.  From  4(a  +  6)  take  2(rt  +  ^).         ^;i5.   2(a  +  ^) 

11.  From  3(a  +  a)  take  (a  +  a.')-  -^1'^^-   2(a  +  x) 

12.  From  9(0*  —  x^)  take  -  2(a2  _  x^). 

Ans.    Il(a2  -  x^) 

13.  From  Qa}  —  loi^  take  —  Za^  +  9Z^'^. 

^1/15.    9rt»  -  24^2 

14.  From  Sa"  —  2^»'*  take  r^*"  —  2/>\  yl;?.9.    2a'". 

15.  From  ^chii^  —  \  tr.ke  4  —  7c'^m^  J«5.  IGc^/i*  — 8. 
10.  From  6am  -f  y  take  3am  —  x  Am.  Sam  -f  a;  -f  y 
17.  From  3aaj  take  3aa;  —  y.  Ans.    -\   y. 


^ 


54  ELEMENTAKY       ALGEBRA. 

-V-  -t  .  _ 

18.  From  —  If  +  Sm  —  8a;  take  —  6/  —  5m  —  2a;  4- 
Sd  4-  8,  Ans.    —/-{-  Bm  —  Gx  —  Sd  —  S. 

19.  From  —  a  _  55  +  '?c  4-  f^  take  ib  ~  c  +  2d  -{-  24 

^n5.    —  a  —  96  +  8c  —  (/  -  2k. 

20.  From  —  3a  +  5  —  8c  +  ^e  —  5/  +  3A  -  Ta;  —  1  ;^./ 
take  Ic  -\-  2a  —  9c  +  Se  —  1x  -\^  If  —  y  —  dl  -f  k. 

Ans.    —  5a  +  ^»  +  c  —  e  —  ]2/  +  3A  —  12y  -f-  3/.     jJl 

21.  From  2a;  —  4a  —  2^  +  5  take  8  —  56  +  a  4   6a;. 

Ans.    —  4a;  —  5a  +  36  —  3. 

22.  From  Za  -\-  h  -^  c  —  d  —  10  take  c  -\-  2a  —  d. 

Ans.  a  4-  6  —  10. 

23.  From  3a  +  6  4-  o  --  c?  —  10  take  6  —  19  4-  3a. 

Ans.   c  —  fZ  4-  9. 

24.  From  a^  4-  36^6  4-  aJ^  _  ahc  take  6^  4-  aJ^  -f  abe.  • 

^n5.    a3  4-  362c  —  b\ 
25    From  12a;  4-  6a  —  46  4-  40  talver46  —  3a  +  4a;  4- 
6ri  —  10 j  A71S.    8a;  +  9a  —  86  —  6^  4-  50. 

26.  From  2a;  —  3a  4-  46  4-  6c  —  50  ta];eV9a  4-  a;  4-  66 

—  6c  -f  40]]  A71S.   X  —  12a  —  26  4-  12c  —  10. 

27.  From  Oa  ~  46  —  12c  +  12a;   take  (2a;  —  8a  -f  46' 

—  Q>c,,  Ans.    14a  —  86  —  6c  4-  10a;. 

38.  In  Algebra,  the  term  difference  does  not  always,  as 
in  Arithmetic,  denote  a  number  less  than  the  minuend.  For, 
if*  from  a  we  subtract  —  6,  the  remainder  will  be  a  4-  6  ; 
and  this  is  numerically  greater  than  a.  We  distinguish 
between  the  two  cases  by  calling  this  result  the  algebraic 
difference. 

88.  lu  Algebra,  as  in  Arithmetic,  docs  the  term  difference  denote  a 
DTimber  less  than  the  minuend  ?  How  are.  the  results  in  the  twc  cases, 
distinguished  from  each_  other  f 


8UBTKACTI0N.  55 

89.  ^V^lcn  a  polynomial  is  to  be  subtracted  from  an  al- 
gebraic quantity,  we  inclose  it  in  a  parenthesis,  place  the 
minus  sh^n  before  it,  and  then  ^Tite  it  ailer  the  minuend 
Thus,  the  expression,  _ 

Ga^  —  (Sab  -  Ih^  +  Ihc), 

indicates  that  the  polynomial,  Zah  —  2^^  ^  25c,  is  to  be 
taken  from  Ca^.  Performing  the  operations  indicated,  by 
the  rule  for  subtraction,  we  have  the  equivalent  expression  : 

T^ie  last  expression  may  be  changed  to  the  former,  by 
changing  the  signs  of  the  last  three  tenns,  inclosing  them  in 
a  parenthesis,  and  prefixing  the  sign  — .     Thus, 

6a*  —  Zdh  +  2^2  _  26c  =  ^cO-  —  (3a6  -  W-  4-  25c). 

In  like  manner  any  polynomial  may  be  transformed,  as  in- 
dicated below : 

=  Ta3  —  8a25  -  {Wc  -  G*^). 
8a3  —  7^2  Ac^  (p_  8a3  —  {W.  —  c  +  ^ 

'   =  8a3  _  7^2  -  (_  c  +  d), 
95^  —  a  -f  3a2  —  c?  =  05^  —  (a  —  Sa^  4.  ^7) 

=  953  -  a  —  (-  Sa^  4-  ^. 

Note. — The  agn  of  every  ^quantity  is  changed  when  it  is 
placed  within  a  parenthesis,  <iBd  also  when  it  is  brought  out. 

4  O.    From  the  preceding  principles,  we  have, 

a  —  (+5)   =  a  —  h\  and 
a  —  (—5)  =  a  -\-  h. 

89.  How  18  the  subtraction  of  a  polynomial  Indicated  ?  Row-  is  this 
lu'jicatod  operation  performed  ?  How  may  the  result  be  again  put  under 
the  first  form  ?     What  is  the  general  rule  in  regard  to  the  parenthesis? 

40.  What  is  the  sign  which  immediately  precedes  a  quantity  called? 
What  is  the  sign  which  precedes  the  parenthesis  called?     What  is  the 


66  ELEMENTARY       ALGEBKA. 

The  sign  immediately  preceding  b  is  called  the  sign  of  tfte 
quantity;  the  sign  precedmg  the  parenthesis  is  called  the 
sign  of  ojjeration  ;  and  the  sign  resulting  from  the  combin- 
ation  of  the  signs,  is  called  the  essential  sign. 

When  the  sign  of  operation  is  different  from  the  sign  of 
tiie  quantity,  the  essential  sign  will  be  —  ;  when  the  sign  of 
operation  is  the  same  as  the  sign  of  the  quantity,  the  esseii* 
tial  sign  will  be  +. 


JMULTIPLICATION. 

41.  1.  If  a  man  earns  a  dollars  in  1  day,  how  much  will 
he  earn  in  6  days? 

Analysis. — In  6  days  he  will  earn  six  times  as  much  as  in 
1  day.  If  he  earns  a  dollars  in  1  day,  in  6  days  he  will  earn 
6a  dollars. 

2.  li"  one  hat  costs  d  dollars,  what  will  9  hats  cost  ? 

Ans.  9f?  dollars. 

3.  If  1  yard  of  cloth  costs  c  dollars,  what  will  10  yards 
cost?  Ans,  10c  dollars. 

4.  K  1  cravat  costs  h  cents,  what  will  40  cost? 

Ans,  40^  cents. 

5.  K  1  pair  of  gloves  costs  b  cents,  what  will  a  pairs 
cost? 

Analysis. — If  1  pair  of  gloves  cost  b  cents,  a  pairs  will 
cost  as  many  times  b  cents  as  there  are  units  in  a :  that  is, 
b  taken  a  times,  or  ah ;  which  denotes  the  product  of  b 
by  a,  or  of  a  by  b, 

resulting  sign  called?  When  the  sign  of  opei ation  is  different  from  the 
sign  of  the  quantity,  what  is  the  essential  sign  ?  When  the  sign  of  ope- 
ration is  the  same  as  the  sign  of  the  quantity,  what  is  the  essential  sign 

41.  What  is  Multiplication  ?  What  is  the  quantity  to  be  multiplied 
called?  What  is  that  called  by  which  it  is  multiplied?  What  is  the 
teeult  called? 


MULTIPLICATION.  57 

Multiplication  is  the  operation  of  findit^g  the  2>roduci 
of  two  quajitities,  • 

The  quantity  to  be  multiplied  is  called  the  Midtiplicand ; 
thai  l)y  which  it  is  multiplied  is  called  the  Multiplier  ;  and 
the  result  is  called  tlie  Product.  The  Multiplier  and  Multi- 
plicand are  called  Factors  of  the  Produrt. 

G.  If'  a  man's  income  is  3a  dollars  a  w  eek,  how  much  \vill 
he  receive  in  Ab  weeks  ? 

Za  X  Ab  =   \2ab. 

If  we  suppose  a  =  4  dollars,  and  b  =  3  weeks,  the  pro- 
duct will  be  144  dollars. 

Note. — It  is  proved  in  Arithmetic  (Davies'  School,  Art.  48. 
University,  Ait.  50),  that  the  product  is  not  altered  by  chajig- 
uig  the  arrangement  of  tlie  factors ;  that  is, 

\2ab  =  a  X  b  X  12   =  b  x  a  x  \2   =  a  x  \2  x  b. 

MTn.TIPLICATION    OF    POSFm'B    MONOMIALS. 

4*i.     Multiply  Sa^ft^  by  2a^b,     We  write, 

3^262  X  2a'^b  =  3  X  2  X  a^  X  a^  X  b^  X  b 
=  3  X  2  a  a  a  ab  b  b\ 

ui  which   a   is  a  factor  4  times,  and   b  a  factor  3  times ; 
hence  (Art.  14), 

Za^b"^  X  2a^b  =  3  x  2a*b^  =  Qa*b\ 

in  which  ice  rmdtiply  the  cot'.ffi,cie7it3  together^  ajid  add  the 
exjx)?i€fits  of  the  like  letters. 

The  j)roduct  of  any  two  positive  monomials  may  be  found 
in  like  manner ;  hence  the 

RI7LB. 

I.    Multiply  the  coefficients  together  for  a  neio  coefficiept: 
II.    Write  after  this  coefficient  all  the  letters  in  both  rnono- 

42^   What  id  the  rule  fot  multiplying  oue  mononiial  by  another? 
3* 


58 


ELEMENTARY      ALGEBRA, 


7)iiaL%  giving  to  each  Utter  an  exponent  equal  to  the  sum  of 
Us  exponents  in  tJie  tico  fadors. 


EXAMPLES, 


Multiply 

by 


^a^hc^  X  lahd"^  =  5QaWc^d\ 
21a^^cd  X  8abc^  =   168a*Z>Vf?. 
4ahc  X  Id/  =  2Sabcd/. 


3a^b 
%a^h 


a^xy 


(5.) 
\2a^x, 
\2xhj 

(8.) 


(C.) 
Qxyz 
ay'iz^ 

Qaxy'^z^ 

(9.) 

87aa;2y 
3PxHf 


•  27a3^»V 


10.  Multiply  5a3^2aj2  j^y  6^^3,6, 

11.  Multiply  lOa^b^c^  by  Yac^Z. 

12.  Multiply  ^QaWc^d^  by  20a52^3(7 

13.  Multiply  5«"'  by  SaS". 

14.  Multiply  Za'^b^  by  6a2J^ 

15.  Multiply  ea'"^*"  by  9a^5^ 

16.  Multiply  5«'"5'»'by  2aPM. 
n.  Multiply  ha'^JP'C^  by  2a^>''c. 

18.  Multiply  Ga^Zf^c"  by  ZaWc^. 

19.  Multiply  l^a^b^cd  by  lla^x^y.     Ans.   240a'^b^cdx^y 

20.  Multiply  14a456(^4y  ]3y  20a'Va;22/.  ^.  280a'^>VfZ»a;V 

21.  Multiply  ^d^b^y^  by  la'^bxy^.  Ans.   BGa'^b^xy^ 

22.  MuUipl)'  Ibaxyz  by  ha^bcd^y"^.  Ans,  315a^bcdx^y^z. 


2Qlab^x''y\ 
Ans.    30«-^ZiVa^ 

yl/i5.  I20a^b^c^d^ 
Ans.    I5a'^'^^b 
Ans.   ISa'^-^^b''-^ 
Ans.   54a'" +  V>'*-^ 
Ans.   10a'"  +  pZ»'»-^r 
J[w5.   10a'"  +  ^Z>'»+V, 
Ans.    18a5^>'"-^2c«  +  2 


M  CLT 


L  I  C  ATIO  N. 


59 


23.  Multiply  OAct^m'^x^yz  Ly  8flr5V.  A,  512a^5Vm«aj<y«. 
■  21.  Multiply  Oa^b^c^d'^  by  12a-*6*c«.  A?is.  lOQa^b^c^d^ 
-25.  Multiply  21  Ca^>^cV8  by  Sa?^»V.     ^;i5.    648a^Z»Vc?» 

20.  Multiply  lOa^'O'c^dyx  by  l^cfb^c^dxhf, 

Ans,   840a»*5^V<fyary 


a  — 

t> 

c 

ac  - 

-  ^ 

8  - 

3 

33 

6 

7     . 

. 

. 

7 

56  - 

21 

~ 

35 

MULTIPLICATION    OP    POLYNOMIALS. 

43.     1.  Multii>ly  a  —  ^  by  c. 

It  ia  required  to  take  the  difference 
between  a  and  b,  c  times ;  or,  to 
take  c,  a  —  b  times. 

As  w«f  can  not  subtract  b  from  c, 
we  begin  by  taking  a^  c  times,  which 
is  ac;  but  this  product  is  too  large 
by  b  taken  c  times,  which  is  be ; 
hence,  the  true  product  is  ac  —  be.  I 

If  a,  b,  and  c,  denote  numbers,  as  a  =  8,   6  =  3,   and 
c  =  7,  the  operation  may  be  written  in  figures. 

Multiply    a  —  b    l)y    c  —  d. 


It  is  required  to  take  a  —  h  as 
many  times  as  there  are  units  in 
o  -  d. 

If  we  take  a  —  b^  c  times,  ve 
have  04:  —  be  \  but  this  product  is 
too  large  hj  a  —  b  taken  d  times. 
But  a  —  b  taken  d  times,  is  ad—db. 
Subtracting  this  product  from  the 
preceding,  by  changing  the  signs  of 
its  terms  (Art.  37),  and  we  have, 

(«  -  «)  X(^  -  "^r  =  «*  - 


a  - 

-b 

c  - 

-  d 

ac- 

-be 

-  ad  -h  bd 

acr- 

-  be  —  ad  +  bd 

S 

-3           = 

5 

7 

-2           = 

5 

56 

-  21 
-16+6 

66 

-37  +  6  = 

25, 

be  - 

-  ad  4.  bd. 

60  ELEMENTARY       ALGEBRA. 

Hence,  we  have  the  foUowmg 

RULEFOKTIIESIGNS. 

I.  Whefi  the  factors  have  like  signs^  the  sign  of  their 
product  will  be  -f  .* 

II.  When  the  factors  have  unlike  signs,  the  sign  of  (heir 
irrodact  will  be  —  : 

Therefore,  we  say  m  Algebraic  language,  that  +  inrJti- 
plied  by  +  ,  or  —  miiltiphecl  by  — ,  gives  -f  ;  —  muki- 
plied  by  -\-     or   -f-   multiplied  by   — ,  gives  — . 

Hence,  for  the  miilti])licalion  of  polynomials,  we  have  the 
foilowmg 

RULE. 

Multiply  every  term  of  the  niultiplicand  by  each,  term  of 
the  multijylier,  observing  that  like  signs  give  -f ,  cin^d  mdike 
nans  —  /  the7i  reduce  the  result  to  its  simplest  form. 

EXAMPLES    IN    WHICH    ALL   THE   TERMS    ARE    PLUS. 

1,  Multiply        ....       'ia^  -{■     Aab  +  b"^ 
by        2a   -t     bb 


6a3  -f    ^d'b-^    lab"^ 
The  product,  after  reducing,  -f  \oaVj-\-  20ab^  -f  bb"^ 

becomes     ....       Ga^  +  2;]aV>+  22a62  ^  5^3^ 

44.  Note. — It  will  be  found  convenient  to  arrayige  the 
terms  of  the  polynomials  with  reference  to  some  letter;  that 
is,  to  write  them  down,  so  that  the  highest  power  of  that 
letter  shall  enter  the  first  term ;  the  next  liigliest,  the 
second  term,  and  so  on  to  the  last  term. 

41.  How  are  the  terms  of  a  polynomial  arranged  with  reference  to  a 
particular  letter  ?  What  is  this  letter  called  ?  It  the  leading  letter  in  the 
multiplicand  and  multiplier  is  the  same,  which  will  be  the  leading  letter 
111  the  product? 


MULTIPLICATION.  61 

The  letter  with  reference  to  which  the  arrangement  is 
made,  is  called  the  leading  letter.  In  tlig  above  example  the 
leading  letter  is  a.  The  leading  letter  of  the  product  will 
always  be  the  saine  as  that  of  the  fActors. 

2.  MiiUii)ly    x^  +  2aa;  -f-  a^    by    a;  -}-  a. 

Ajis,  x3  +  3ax2  +  3a2aj  +  a^ 

3.  Multiply    x^  -\-  y^    by    x  -\-  y, 

Ans.  aj<  +  a^^  -f  aJ^y  +  y*. 

'4.  Multiply    3a62_^6(/V    by    Sa^^^  +  3c/V. 

Ans.  ^d'b^.+  lld'h^c^  +  18aV. 

5.  Multiply    d^ly^  -f  ed    by    a  -\-  b. 

/  Ans.  a^b^  +  (K^^d  +  a^i^  +  bcH. 

6.  Multiply    3aar^  +  Oo^^  -f-  cc/*    by    6aV. 

^7W.  18a\'2{c2  +  54a  V63  +  ^a?-L^d'>. 

7.  Multiply    64aV  +  lla'^x  +  9ai    by    %a\d. 

Ajis.  5\2a^cdx^  -f  216a*c(iB   f-  12a*b€d 

0.  Multiply    a^  +  J^a^aj  4-  Gax^  +  x3    by    a  +  x. 
^  Ans.  a*  +  4a^x  +  Ca222  ^  4^,5^53  4.  ^^ 

9.     Multiply        aj2    _|_     y2        y^y        g.     _|.     y^ 

-4?i5.  ar'  +  a;y2  4-  3^2  y  _j_  ys^ 

10.  Multiply  a:*  +  xif  -)-  7«aj    by    ax  -\-  box. 

A?2S.  Qax^  -f  6c/xy  ^  42a2aj> 

11.  Multiply    a^  +  Oa^j  4.  zab^  4.  53    i,y    ^  ^  ^^ 

Ans.  a*  +  4a36  +  Qa^b^  -r  4a^3  +  &*. 

12.  Multiply    ar*  +  a^^y  +  ary^  4-  y^    by    a;  4-  y. 

Ans.  X*  4-  2ar''y  +  2x-y^  -f  2a^5  ^.  y4^ 

13.  Multiply    ar3  4-  2x2  4-  a:  +  3    i,y    3aj  4-  1. 

A7is.   Xr\  4    7.r3  -f  'yx"^'  4-   lOx  4-  3, 


62  ELEMENTARY        ALGEBRA. 

GENERAL      EXAMPLES. 

I.  Multiply    .     .• 2ax  —    2,ah 

by 3a;    —       h. 

The  product ^ax^—    ^ahx 

becomes  after —    ^abx  +  3ffJ^ 

reducing   ...  ,     .     .     .    .^ax^—  Wahx  +  Zah"^. 

2.  Multiply    a*  —  W    hj    a  —  b.      . 

A?is.  a^  —  lab^  —  a^h  -^  25*. 

3.  Multiply    {«2  _  3^.  __  7    ])y    a;  _  2. 

An?,  x^  —  5x'^  —  jc  +  14. 

4.  Multiply    3a2  —  5ab  +  2b^    by    a^  —  lab. 

Ans.  3a*  —  2Qw^b  +  Zla'^b'^  —  Uab\ 

5.  Multiply    b^  +  b^  -\-  b^    by    b''-  —  1.      A71S.  b^  -  b\ 

6.  Multiply  £c*—  2x^y-{-^x^y'^—Qx\f-\-  IGy*  by  a;+  2?/. 

Ans.  x^  -\-  32y\ 
1.  Multiply    4a!^  —  2y    by    2?/.  A^is.  Qx^y  —  Ay"^. 

8.  Multiply    2x  +  4y    by    2x  —  Ay.     Ans.  Ax^  —  l&y\ 

9.  Multiply    x^  +  x^y  -f-  cc?/^  -f  ^3    i,j    y^  _  y^ 

Ans.  X*  —  y*. 
.  10.  Multiply    a;2  _{_  jf-y  _|_  y2    j^y    2^2  _  ^.y  _l_  ^2^ 

ylws.  x^  4-  a;2y2  _j_  2/4 

II.  Multiply   2a^  —  3«£c  +  Ax"^  by   5«2  _  g^,^  _  2^32. 

u4ws.  10a*  —  2la^x  4-  34a2cc2  _  iQax^  —  Sx\ 
12.  Multiply   3a;2  _  2i:cy  +  5    by   a;^  +  2a:y  —  3. 

Ans.  8a?*  +  Ax^y  —  4a;2  —  4a;y  +  IQxy  —  15. 


13.  Multiply    dx^  H-  2.'c"i/2  +  Sy^   by   2a;3  —  Scc^y^  ^  5^3 

j  6a;«  -  5a;V^  —  6ccy  +  6iry  -f 
^^*   I  ISccV  —  ^^V  +  lOtc'y^  -f  15y\ 

14.  Multiply    Sax  —  Gab  —  c  by   2ax  +  ab  -\-  c. 
A71S.  IGa^a;^  —  Aa'^bx  —  ^a^-b"^  -h  Gac^.  —  labv,  —  c'^. 


DIVISION.  DO 

16.  Mullfply   3a2  -   5^2  -f  Sc^   by   a^  -  b\ 

Ans.  d(i*  -  Sarh-  +  3aV  +  5fi*  -  W(^. 

16.  3a2  _  tihd  4-    C/" 

iVo.rcd.  -  15a*  +  Z1a^bd-2^a\'f-2QbhP^A^hcdf-^c^P 

17.  Multiply  arx  —  a'^b'^  by   a^a;". 

16.  Multiply  «*"+  i"   by  a*"—  ^>\         -4/i3.  a^**  —  li^\ 
19.  Multiply  rt-  +  i"   by   «»"  +  6". 


DIVISION. 

4, "5,  Dn'isK^N  is  the  operation  of  finding  from  two  quan- 
tities a  third,  which  being  multiplied  by  the  second,  will 
produce  the  first. 

Tlie  first  is  called  the  Dividend^  the  second  the  Divisor^ 
and  the  third, the  Quotient. 

Division  is  the  converse  of  Multiplication.  In  it^  we  have 
given  the  product  and  one  factor,  to  find  the  other.  The 
rules  for  Division  are  just  the  converse  of  those  for  Multi- 
plication. 

To  divide  one  monomial  by  another, 

46.  Divide  72a'  by  8a^.  The  division  is  indicated, 
thus : 

8a3  * 

Tlic  quotient  must  be  such  a  monomial,  as,  being  multiplied 
by  the  divisor^  will  give  the  dividend.     Hence,  the  coefficient 

46.  Wh\t  is  division  ?    What  is  the  first  quantity  called?    The  9ec0r.fl? 
llic  third  •     What  is  given  in  division?     What  is  required  ? 
40.  What  is  the  rule  for  the  dlviHion  of  moiwmiial}*? 


64  ELEMENTAKV       ALGEBRA. 

of  the  quotient  must  be  9,  and  the  literal  part  (i^ ;  for  tliese 
quantities  multiplied  by  %a?  will  give  72a^.     Hence, 

The  coefficient  9  is  obtained  by  dividing  72  by  8;  and 
tlie  literal  part  is  found  by  giving  to  «,  an  exponent  equal 
to  6  minus  3. 

Hence,  for  dividing  one  monomial  by  another,  we  have 
the  following 

RULE. 

I.  Divide  the  coefficient  of  the  dividend  hy  the  coefficient 
of  th,e  divisor^  for  a  new  coefficient  : 

II.  Aft  at'  this  coefficient  icrlte  all  the  letters  of  the  dividend^ 
giving  to  each  an  exponent  equal  to  the  excess  of  its  expo- 
ponent  in  the  dividend  over  that  in  the  divisor. 

SIGNS    IN    DIVISION. 

47.  Since  the  Quotient  multiplied  by  the  Divisor  must 
l)roduce  the  Dividend :  and,  shice  the  product  of  two  factors 
having  the  same  sign  will  be  -f  ;  and  the  product  of  two 
factors  having  different  signs  will  be   —  ;  we  conclude: 

1.  When  the  signs  of  the  dividend  and  divisor  are  Uke, 
the  sign  of  the  quotient  Avill  be  -{-. 

2.  When  the  signs  of  the  dividend  and  divisor  are  unlike, 
the  sign  of  the  quotient  will  be  — .  Again,  for  brevity,  we 
say, 

-f  divided  by  +,  and  —  divided  by  — ,  give  +  ; 
—  divided  by  -f ,  and  -f  divided  by  — ,  give  — , 
4-  «^  ,    ,  —ah 

4-  a    ~  '  —  a    ~ 


47.  What  is  the  rule  for  the  pigiw,  in  divisiou  ? 


BI  V  I  8I0N. 


65 


EXAMPLES. 


(1.) 


=    I-  2a^b, 


(2) 


(3.) 


-  24a^ 
4-    3adc 


=  -  8a\ 


—    5u^x 


32a^b'^x^ 


8a'b^ 


5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
26. 


Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divnde 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Di\nde 
Divide 


15aa^y3  by  —  Say. 

Siab^x  by  I2b\ 

—  3Ga^6V  by  da^b^c, 

—  Qda'^b^x^  by  11  a^b^x\ 
lOSjc^y^s^  by  54a:'^z. 
64ic''y*2«  by  —  16a;*'j^s*. 

—  9Ga^5V  by  I2a^bc. 

—  '3Qa*b^d*  by  2a^b\l. 

—  Qia^b*c^  by  32a**c. 
\28a^3f"y'  by  IBoary*.  ' 

—  2bQa'b^c\V  by  16a^6c«. 
200a^wi2/i2  by  —  50a''m?i. 
300aryz2  by  COajy^^, 
2na^b'^c^  by  -^  9a*c. 
G4ayz®  by  32a2/V. 

—  88a^^»6c8  by  11  a^^^c*. 
Ila^y^z}  by  —  llaV^*- 
8\a^b'^c^d  by  —  42a*b'^c^d. 

—  8%a^b\^  by  8a''^.«c«. 
1Gj:2  i^y   _  8iB. 

—  88a"6^  by  lla"»6. 


(4.) 
=   —  4<2aj*. 

Ana,    —  5a;2y2^ 
-^?w.    7</^a% 

A  718.    —  Aab^c. 

Ans.    —  dab^x. 

Ans.    2xyV. 

Ans.    —  4ipy2. 


-4/45. 


8a'^6\ 


^7W.    —  Idabd^. 

Ans.    —  2ab^c\ 

Ans.   8«^ar''y\  ^ 

Ans.  —  leab^chP. 

A?is,    —  Aanin. 

Ans.   bx^y-z. 

Ans,   —  da^bc. 

Ans,   2a^y2. 

Ans,    —  &d^b^c\ 

Ans,   —  7. 

Afis,    —  2, 

Ans,    —  lla^. 

Ans,    —  2x. 

Afis,    —  8a*»~'"& 


66  ELEMENTARY     ALGEBRA. 

26.  Diyide  77«'"2''*  by  -  lla'd\  Ans.  -ndT-^lr-^ 

27.  Diyide  84a'5'"  by  ^%a%\  Ans.  2fi«-"Z'"'-^ 

28.  Divide  -  '^^aJ'lf  by  WlT,  Ans.  -  llfl^"*'^^-'". 

29.  Divide  ^Ul^  by  A^O'l^  Ans.  2a'-''h^-K 

30.  Divide  16Saff  by  12^^"?^  ^?^5.  14af-Y-'". 
3L  Divide  256^5 V  by  lea^S^c^.  ^ws.  lea^-^S^^V--^. 

MONOMIAL   FKAOnONS. 

48.  It  foUo^vs  from  the  preceding  rules,  that  the  exact 
division  of  monomials  will  be  impossible : 

1st.  When  the  coefficient  of  the  dividend  is  not  exactly 
divisible  by  that  of  the  divisor. 

2d.  Wlien  the  exponent  of  the  same  letter  is  greater  in 
the  divisor  than  in  the  dividend. 

3d.  When  tlie  divisor  contains  one  or  more  letters  not 
found  m  the  dividend. 

In  either  case,  the  quotient  will  be  expressed  by  a  fraction. 
A  fraction  is  said  to  be  in  its  simplest  form^  when  the 
numerator  and  denominator  do  not  contain  a  common  factor. 
For  example,  Xla^lP-cd^  divided  by  Sa^Jc^,  gives 

"8a2^  ' 

which  may  be  reduced  by  dividing  the  numerator  and  de- 
nominator by  the  common  factors,  4,  a^^  j^  and  c,  giving 


Also, 


loa'b^d*        Zb^d 


15.  TTnder  what  circumstances  will  the  division  of  monomials  be  im- 
popyible  ?  IIow  will  the  quantities  then  be  expressed  ?  How  is  a  mono 
mial  fraction  reduced  to  its  simplest  form? 


DIVISION.  67 

Hence,  for  the  reduction  of  a  monomial  fi-action  to  its  sira 
plest  fonu,  we  have  the  following 

EXILE. 

Suppress  every  factor,  lohether  7iumerical  or  literal^  that 
U  common  to  both  terms  of  Vie  fraction  :  the  result  will  be 
the  reduced  fraction  sought, 

EXAMPLES. 

(1.)  (2.) 

48a^yc(F    _  4a^_       ,  Z1a¥c\l    _  3Wc  , 
aCa^iVrfe  ~  Zbce'  ^^      6a'hcHP  ~    ed^d' 

(3.)  .      (4.) 

la^b  1  ,   4('J2         2a 

5.  Divide  iOa^b^c^  by  Ua^bc\  Arts,  — -. 

6.  Divide  6am?i  by  Zahc,  Ans.  —. —  • 

•'  he 

*l.  Divide  \%a^l)^mn^  by  \1a^h^cd.  Am,  7— r;,— ,• 

'I  ( i^c^d 
8.  Divide  28a*^VcZ^  by  IQah^cdPm,  Ans,  —,, — 

^  ''  Ab^m 

f* 

9   Divide  na'^c^b'^  by  12aV6\?.  Ans,  -^£-,- 

—  "^  a^c^bd 

^.  .,  «,e  ,  n,. -,  ^        Aa^bxmn 

10.  Divade  lOOa^o-^XTW/i  by  25a3J*c/.  Ans,  ^ • 

11.  Divide  ^Qa^b^cW  by  Iha^cxy,         Ans.  — ^^ ^» 

'— •  ^      V  ./  2o»y 

12.  Divide  8om^n}fxhj^  by  ISam^n/l       -4??«.  -x— r" 

13.  Divide  I27(ra:y  by  16c/»a;*y*.  Ans,  ^-—^ 


OS  ELEMENTARY      ALGEBRA. 

49,  In  dividing  monomials,  it  often  happens  that  the 
exponents  of  tlie  same  letter,  in  the  dividend  and  divisor, 
are  equal;  in  wliich  case  that  letter  may  not  appear  in  the 
quotient.  It  might,  however,  be  retamed  by  givmg  to  it  the 
exponent  0. 

If  we  liaA^e  expressions  of  the  form 

a      a?      a^      «>      a^ 

a'    a^'    a3>    ^»    ^^    *^c., 

and  apply  the  rule  for  the  exponents,  we  shall  have, 

a  a?  '    a^  ' 

But  since  any  quantity  divided  by  itself  is  equal  to  1,  it  fol- 
lows that, 

-  .=  ao  ^  1,   ^  r=  a2-2  ^  f^o  ^  1    &c. ; 
or,  finally,   if  we  designate  the  exponent  by  m,  we  have, . 
—  =  cr""*  =  a^  =.1;   that  is. 

The  0  po'uber  of  any  quantity  is  equal  to  1  :  therefore, 
Any  quantity  may  he  retained  in  a  term^  or  introduced 
bito  a  tenn^  by  giving  it  the  exponent  0. 

EXAMPLES. 

1.  Divide  Ga^^V  by  laW. 

— ^y-    =     3^2-2^2-2^4    ^     3^050^4    ^     3^, 

2a-b^ 

2.  Divide  Sa^J^c*  by  —  ^a'^h'^c.   Ans.  —  2a"Z^"6'*  —  —  2c'*. 

3.  Divide    -  ?>1mVx^y'^  by  Amhi^xy. 

Ans.   —  ^m^n^xy   —   —  8,Ty. 


49.  When  the  exponents  of  the  same  letter  in  the  dividend  and  divisor 
are  equal,  what  takes  place  ?  May  the  letter  still  be  retained  ?  With 
what  exponent?     What  is  the  zero  power  of  any  quantity  equal  to? 


DIVISION.  69 

4.  Dii-ide  —  96a<JV"  l)y  —  24a*6*.   Ans.  4a"5**o*'»  =  4c" 
6.  Introduce  a,  as  a  factor,  into  CiY\  Ans.  6cWc*, 

6.  Introduce  aby  as  factors,  into  OcW".      A71S.  9a^'b^'c-\i'*, 

7.  Litroduce  a^c,  as  factors,  into  S(I*/*>.     A.  ^a%^c^d\f'^. 

50.  WTien  the  exponent  of  any  letter  is  greater  in  the 
divisor  tlian  it  is  in  the  dividend,  the  exponent  of  that  letter 
in  the  quotient  may  he  written  with  a  negative  sign.     Thus, 

;-^,    by  the  rale; 


b  X  a-^  =  -^, 


I         ,         «^ 

hence, 

•-=i 

Since, 

a  -  3  =  —  ,     we  have. 

a' 

that  is,  a  in  tlie  numerator,  wit!)  a  negative  exponent,  is 
equal  to  a  in  the  denominator,  with  an  equal  i)08itive  ex- 
ponent; hence, 

Any  quantity  having  a  negative  exponent,  is  equal  to  tfi^ 
reciprocal  of  the  same  quantity  with  an  equal  positive  ex- 
ponent. 

Ilence,  also, 

Any  factor  may  be  transferred  from  the  denominator  to 
the  numerator  of  a  fraction,  or  the  reverse,  by  changing  the 
sign  of  its  exponent.  ^ 

EXAMPLES.  ^5^ 

1.  Divide  Z2a^bc  by  IGa^R 

,        32a2^>c        ^   _-._,  2c 

^^-  16^^  =  '^     ^    ''  =  a^h- 

50.  When  the  exponent  of  any  letter  in  the  divisor  is  greater  than  in 
the  dividend,  how  may  the  exponent  of  that  letter  be  written  in  the  quo- 
tient ?  What  is  a  quantity  with  a  negative  exponent  equal  to  ?  How 
may  a  factor  be  transferred  from  the  numerator  to  the  denominator  of  a 
fraction  ? 


70  ELEMENTARY      ALGEBRA. 

3.  Reduce    ^,   f  „  •  ^;i5.  ,  or  — z 

I  blx^y^  3  3a;^ 

f      4.  Ill  5ai/~^x~'^,  get  rid  of  the  negative  exponents. 

A  ^^ 

^'  Iji       _3t_5?  get  rid  of  the  negative  exponents. 

Aois. 


3£C2. 

6.  Li         _^-  _g  _   ,  get  rid  of  the  negative  exponents. 

da  c  cr 

7.  Reduce     ^^   „,_, —      Ans. ,  or  --^^-  • 

9 

8.  Reduce  l^a^Jy^  -i-  8a^&^  ^^s.  9a~i5~\  or  — r. 

15(^  —  4^6^—1 

9.  In  -2h-\    '  S^^  ^^^  ^^  ^^®  negative  exponents. 

Ans.  -^' 

10.  Reduce z^ — =— •  Ans.  Sab^c^, 

—  ba~^b~^ 

To  divide  a  polynomial  by  a  monomial. 

51.     To  divide  a  polynomial  by  a  monomial : 

Divide  each  term  of  the  dividend^  separately^  by  the 

divisor  ;  the  algebraic  sum  of  the  quotients  will  be  the  quo- 

ilent  sought, 

EXAMPLES. 

1.  Divide  ZaW  -  a  by  a.  Ans.  ZaV^  —  1. 


51,  How  do  you  divide  a  polynomkil  by  a  monomial? 


DIVISION.  71 

2.  Divide  ba^b"^  —  2oa*b^  by  5a'b\  Ans,   1  —  5a, 

3.  Divide  35arb^  —  25ab  by  —  5ub.     Ans,    —  lab  +  5 
i.  Divide  lOab  —  I5ac  by  5a.  A713,   2b    -  3c. 

5.  Divide  6a^  —  8aa;  -f  4a2y  by  'ia. 

^w*.   db  —  4x  -i-  2ay, 

6.  Di\nde  —  ISaa;^  +  coj^  by  —  3a;.      A7\s,   5ax  —  2x\ 

7.  Divide  -  2\xy^  -f  Sba^b^i/  -  Ic^j  by  -  7y. 

yl;!.*?.    3iry  —  ha'^b^  +   c*. 

8.  Divide  iOa^b*  +  8a*6'  -  32a^i*c*  by  Qa*b\ 

Ans.  on*  -\-  b^  —  4c* 

DIVISION     OF    rOLYNOinALS. 

52.     1.  Divide   —  2a  +  Ca^  _  8  by  2  +  2a. 

Dividend.      Divisor. 
6a2  —  2a  —  8  |  2a.  +  2 
ba^  4-  6a  3a  —  4     Quotient 

—  8a  —  8 


—  8a  —  8 


Bemainder, 


We  first  arrange  the  dividend  and  divisor  with  reference 
to  a  (Alt.  44),  placing  the  dinsor  on  the  left  of  the  dividend. 
Divide  the  first  term  of  the  dividend  by  the  first  term  of 
the  divisor;  the  result  will  be  the  first  term  of  the  quotient, 
which,  for  convenience,  we  place  under  the  divisor.  The 
product  of  the  divisor  by  this  terra  (Ba^  -t-  Ca),  being  sub- 
tracted from  the  dividend,  leaves  a  new  dividend,  which  may 
ne  treated  in  the  same  way  as  the  original  one,  and  so  on  tn 
the  end  of  the  operation. 

52.  What  is  the  rule  for  dividing  one  polynomial  by  another  ?  Wlicn 
if  the  division  exact  t    When  is  it  not  exact  f 


72  ELEMENTARY       A  L  O  E  B  E  A  . 

Since  all  similar  cases  may  be  treated  in  the  same  way,  we 
have,  for  the  division  of  polynomials,  the  following 

E  ULE. 

I.  Arraiige  the  dimdend  and  divisor  with  reference  to  the 
same  letter: 

n.  Divide  the  first  term  of  the  dividend  hy  the  first  term 
cf  the  divisor^  for  the  first  term  of  the  quotient.  Multiply 
the  divisor  hy  this  term  of  the  quotmit,  and  subtract  the 
product  from  the  dividend: 

ni.  Divide  the  first  term  of  the  remainder  hy  the  first 
terrn  of  the  divisor^  for  the  second  term  of  the  quotient, 
Mxdtlply  the  divisor  hy  this  term,  and  subtract  the  product 
froTR  the  first  remahider,  and  so  on: 

rV.  Continue  the  operation,  until  a  remainder  is  found 
equcd  to  0,  or  one  ichose  first  term  is  not  divisible  by  that 
of  the  divisor. 

Note. — 1.  When  a  remainder  is  found  equal  to  0,  the 
division  is  exact. 

2.  "Wlien  a  remainder  is  found  whose  first  term  is  not 
divisible  by  the  first  term  of  the  divisor,  the  exact  division 
is  impossible.  In  that  case,  write  the  last  remainder  after 
the  quotient  found,  placing  the  di\'isor  under  it,  in  the  fonn 
of  a  fraction. 

SECOND     EXAMPLE. 

Let  it  be  required  to  divide 
51a'^^>2.4.  loa*  -  4.Qa^b  -  155^+  ^ab"^    hy    ^ah  -  Sa^  -f-  zh\ 

We  first  arrange  the  dividend  and  di\^sor  vnth  reference 
to  a. 


DIVISION. 


73 


IXuidend, 
1  Oa*  -  4  M^b  -h  5 1  a^h'^  +  Aab^  -  1 55* 
4  10a*—    ^a^b-    Qd'b^ 


Divisor, 
5a' +  4ad  4-  86* 


—  40a^b-^51a'^b'+  4ab^-^l5b' 
~  iOa^  +  32a^5^4-24a5^ 

25a^^—20ab^^l5b^ 
2oa^b^—20ab^—l5b^ 


—  2a^+  Sab  —  56» 


(3.) 
aj«  4-  Jc3y  4-  a;2y  +  ay2  _  2y  |  g  4-  y 
jc*  4-  ar'y 


0^4-  scy    ' 


4-  x^y  -f  a?/* 
4-  x^y  4  ccy^ 


2y 


-  2y 

Here  the  division  is  not  exact,  and  the  quotient  is  frac- 
donoL 


(4.) 


1  4-    a 
1  -    a 


1  4-  2a  -I-  2a2  4-  Sa^  + ,  &c. 


-f  2a 

4-  2a  —  2a^ 

4-  2a2 

4-  2a^  —  2a^ 
4-  2a3 

In  this  example  the  operation  docs  not  terminate.     It  may 
be  continued  to  any  extent. 


BXAMP'^BS. 

1.  Divide  a'  -I-  2cw;  +  x^  by  a  4-  «.  Ana.  a  -^  x, 

2.  Divide  a'  —  3a^y  4  3ay2  —  y^  by  a  —  y, 

Ajis.  a*  —  2ay  4-  y*. 


74  ELEMENTARY       ALGEBKA. 

a.  Divide  24a'^6  —  \2aW''  —  eab   by    —  Cab. 

Ans.  —  4a  4-  2a-cb  -f  1.    pi 
4.  Divide    6a^  —  96    by    Sx  —  6. 

Ans.  2x^  4-  4a;^  4-  8x  -f-  16. 

^-     6.  Divide     a^  —  5a*a;  4-  lOa-^x^  —  lOa'^ic^  4-  5a2?*  ~  x^ 

by   a^  —  2gkb  4-  a^.  -4/is.  a^  —  'Sa'^x  +  dax^  —  vK 

/     6.  Divide  48ic3  —  76«ic^  —  64a2a;  4-  105a^   by    2ic  —  3a. 

Ans.  24x^  —  2ax  —  ^5a\ 

'7    Divide    y^  —  Sy^x"^  +  Si/'^x'^  —  x^    by    y'-^  —  Sy'^x  4- 

3yx^  —  x^.  Ans.  y^  +  dy'^x  +  Zyx^  4-  x^, 

^     8.  Divide   64a^^>«>  —  2bd%''  by    8a-6^  4-  ^ab\ 

Ans.  8a^b'^  —  bab^, 
9.  Divide   Ca^ -\- 2^a^b  +  22ab'^  +  bb'   by    3«-^4- 4^/^4-R 

^l?^6\  2a  +  56. 
^^    10.  Divide   6aaj«  4-  Oc/ir^y^  4-  42^/2,^.2   by^aic^^js^^ 

A') IS.  x^  +  crv/  4-  'J(}X.^ 

11.  Divide  —  15a^  4-  ^Icfbd  —  2^da\f  —  2()b'-d'  4-  446('4/ 
'    -  8cy^   ])y    3t<2  _  5^ J  ^  (.y^;      ^y^^s-.   -  5a^  4-  4^^/  -  8c/. 

12.  Divide  oc*  4-  aY^  4-  y*   by   «''  —  x,t/  +  t/^. 

J.y/6'.  £c^  +  a;y/  4-  y'^, 

13.  Divide  x*  —  y*   by   £c  —  y. 

^/is.  aj^  4-  x^y  4-  £cy'^  +  y\ 

14.  Divide   2a'-  8a^b''-\-  3aV4-  ^b*-  36V   by    a2_  ^2 

Ans.  Sa^  —  56^  4-  3c". 

15.  Divide   6^«  -  hxHf' -  Cx^y^-k-  Cx^y'^  IbxY—  9a;V 
-f  10a;'y  4-  15y'   by    2x^  +  235^2/^  +  32/2. 

Ans.  2x^  -  ^x^y'^  4-  5y^ 

16.  Divide    —  c^-f-  16a"^a;''^—  nabc  —  4ry%/;  —  6^/2*24  6a«z 
by   8aa;  —  6a6  —  c.  Ans.  2ax  4-  «6  -f  c. 

lY.  Divide    3x^  4-  ^o^'y  —  4a;''  —  ^x'y'-  4    lOa-y   -  15    b> 
2ar</  4-  x"^  —  3.  Ans.   •?>x^  —  2iKy  t  5    ' 


DIVISION.  75 

18.  Divide   a^  4-  32?/'   by   x  -\-  2y. 

Ans.  X*  —  2»«y  +  4a^y2  -  Sjcy^  ^  jgyH^ 

19.  Divide  3a*  —  2^a^b  —  Wab'^  +  37a2^»'''  by  26^  —  6a6 
^  8a^  ^n«.  a^  —  7a^. 

20.  Divide    a*  -  6*    by    a^  +  a^i  +  ab^  +  i^. 

-4y<tf.  a  —  b 

21.  Divide    x^  —  Sai^y  -\-  nf    by    x  -\-  y. 

^         ^        «  +  y 

22.  Divide     1  +  2a  by  1  —  a  —  a^. 

Am.  1  -f  3a  -f  4a»  +  7rt '  +  ,  &c.    ;^ 


76  ELEMENTARY       ALGEBSA, 


CHAPTER   m. 

USEFUL  FORMULAS.     FACTORING.     GREATEST  COMMON  DIVISOll. 
LEAST  COMMON   MULTIPLE. 

USEFUL    FOE:^^DXAS. 

53.  A  Formula  is  an  algebraic  expression  of  a  general 
rule,  or  principle. 

Formulas  serve  to  shorten  algebraic  operations,  and  are 
also  of  much  use  in  the  operation  of  factoring.  When  trans- 
lated into  common  language,  they  give  rise  to  practical  rules. 

The  verification  of  the  following  formulas  afibrds  addi- 
tional exercises  in  Multiplication  and  Division. 

(1.) 

54.  To  form  the  square  of  a  +  5,  we  have, 

(a  4-  by  =  {a  A-  b)  (a  -f-  ^)  =  a'  4-  2ab  -f  b\ 
That  is, 

The  square  of  the  sum  of  any  two  quantities  is  equal  to 
the  square  of  the  firsts  plus  twice  the  product  of  thejirst  by 
the  seco7id,  plus  the  square  of  the  second, 

1.  Find  the  square  of  2a  -f  Sp.     We  have  from  the  rule, 
(2a  +  35)2  =  4a2  -j-  Uab  -h  9b\ 

58.  What  is  a  formula?    What  are  the  uses  of  formulas  ? 

64.  What  is  the  square  of  the  Bxihx  of  two  quantities  equal  to  ? 


USEFCL      K0RMULA6.  77 

•  2.  Find  the  square  of  5ab  -\-  Sac. 

Ans,  2oa^b^  +  SOa^bc  +  9a^c^, 

3.  rind  the  square  of  Sa^  -f  Sa^b, 

Ans.  25a*  {-  80a*b  -4   64a«^. 

4.  Find  the  square  of  Gax  +  Qahi"^. 

Ans.  dQa'^x^  -h  lOSa^a:^  ^  81a*aj* 

(2.) 

55.  To  form  the  square  of  a  difference,  a  —  b,  we  have,^ 
(a  -  6)2  =  (a  -  ^»)  (a  -  5)  =  a2  __  2ab  +  b\ 

That  is, 

The  square  of  the  difference  of  any  two  quantities  is 
eqttal  to  the  square  of  the  firsts  nmius  twice  the  product  of 
the  first  by  tfte  second^  />^w5  the  square  of  the  second. 

1.  Find  the  square  of  2a  —  b.     We  have, 

(2a  ~  by  =  4a2  —  Aab  +  b'^- 

2.  Find  the  square  of  4ac  —  be, 

Ans.  16aV  —  ^abc^  +  b'^c^. 
8.  Find  the  square  of  la^b"^  —  12aR 

Ans.  40a*b*  -  USa^^  -f  144a2^»«. 

(3.) 

56.  Multiply  a  -{-  b  hj  a  —  b.     We  have, 

(a  -f-  6)  X  (a  -  5)  =  a2  —  ^>2,     Hence, 

77/e  .«v7/?  c>/*  ^?<?o  quantities,  multiplied  by  their  difference^ 
id  equal  to  the  difference  of  their  squares. 

1.  Multiply  2c  -\-  b'  by  2c  —  b.  Ans.  Ac"^  —  .6^ 

2.  Multii)ly  9«c  -H  3Jc  by  9(/c  —  3/>c. 

^7?.?.  81  a V  —  9*V 

66.  What  is  the  square  of  the  difForcnce  of  two  quantities  equal  to? 
66.  What  is  the  sum  of  two  quiintiticf)  uultiplicd  hy  their  differcnuo 
eqoal  to  ? 


78  ELEMENTARY       ALGEBRA. 

3.  Multiply  8a3  -f-  ^ah"^  by  Sa^  _  ^obK 

Ans.  UaP  -  i-:)a,W 

57  •    IVrUtiply  a^  ^-  ah  +  6^  by  a  -  b.     We  have, 
{a?  -{-  ab  ^  b"")  (a  -  b)   =   a'  -  b\ 

(5.) 

58.  Multiply  a^  —  ab  -\-  b"^  hy  a  -\-  b.     We  have, 
(a2  -  ab  ^  62)  {a  ^  b)  ^  a?  -^  bK 

(6.) 

59.  Multiply  together,  a  +  J,  a  —  b^  and  a?  -(-  5'. 
We  have,         c»/    ^  ^ 

(a    +    ^>)    (a    -    J)    (a2   +    ^2)     ::::,     ej4  _    ^4^ 

60.  Since  every  product  is  divisible  by  any  of  its  factors, 
each  formula  establishes  the  principle  set  opposite  its  number. 

1.  The  Slim  of  the  squares  of  any  two  quantities^  plus 
ttoice  their  product,  is  divisible  by  their  sum. 

2.  The  sum  of  the  squares  of  any  two  quantities,  minus 
twice  their  product,  is  divisible  by  the  differ e7ice  of  the 
quantities. 

3.  The  difference  of  the  squares  of  any  two  quantities 
is  divisible  by  the  sum  of  the  quantities,  and  also  by  their 
difference. 

4.  The  difference  of  the  C7d)es  of  any  two  quantities  is 
divisible  by  the  difference  of  the  quantities ;  also,  by  the 
sum  of  their  squares,  plus  their  product. 

5.  The  sum  of  the  cubes  of  any  two  quantities  is  divisi 

60.  By  what  is  any  product  divisible  ?  By  applying  this  principle,  wha^ 
follows  from  Formula  (1 )  ?  What  from  (2)?  What  from  (3)  ?  What  from 
{4:)'i>     What  from  (5)?     What  from  (6)? 


FACTORING.  7JI 

We  by  the  sum  of  the  quantities  ;  also.,  by  the  sum  of  the.ii 
squares  minus  th^ir  product, 

G.  The  difference  between  the  fourth  powers  of  any  t^no 
quantities  is  divisible  by  the  sum  of  the  quantities^  by  their 
differetice^  lyy  the  sum  of  their  sqv^reSy  and  by  the  dif 
jcTcnce  of  their  squares. 


FACTORING. 

61.  Factoring  is  the  operation  of  resolving  a  quantity 
into  factors.  The  principles  eni])loy(Hl  are  the  converse  of 
those  of  INIultiplication.  The  operations  of  factoring  a>e 
performed  by  inspection. 

1.  Wliat  are  the  factors  of  the  pol}'nomial 

ac  4-  ob  +  dd. 

We  see,  by  inspection,  that  «  is  a  common  factor  of  all 
the  terms;  lience,  it  may  be  ])laced  without  a  parentliesia, 
ajid  the  otner  pans  within  ;  thus  : 

ac  -^  ah  •\-  ad  —  ^c  +  ^  +  d\ 

2.  Find  the  factors  of  the  polyL  nmial   a'^b'^  -f-  fi'^d  —  a\f. 

Ans.  a'^{b'^  -f  d  —  f), 

3.  Find  tne  factors  of  the  polynomial  ^a^b  —  Ga'^b^  -I-  b^d. 

A?is.  b{^a^  -  Qfi^b  -f  bd). 

4.  Find  the  factors  of  Sa^b  —  da'^c  ~  \Sn^jy. 

Ans,  Sd^(b  —  t^c  —  Qxy), 
6.  Find  the  factors  of  Qa^cx  —  1  Sacoc^  +  la&y  —  aoa^". 
Ans.  2ac{4ax  —  9x^  +  c*y  —  15^/V**). 
6.  Fjictor  300**2^  _  eaWd^  -f-  1  Sal&^c^. 

Ans.  Qa''b'^{5nr  _  ,J*  ^  ^c^). 
^      7.  Factor  I2c*bd^  —  }5c\l*  —  6c^d^f 

Ans.  3c2(/3(4c2^  _  5cd  -  2/). 

CI    Whatip  faotorinel' 


80  ELEMENTARY       ALGEBRA. 

8.  Factor  \5a^hcf  —  lOabc"^  —  Ihdbcd. 

Ans.  babc{Za^f—  2S^  -  bd). 

62.  When  two  terms  of  a  trinomial  are  squares,  and 
positive,  and  the  third  term  is  equal  to  twice  the  product  of 
their  square  roots,  the  trmomial  may  be  resolved  into  factors 
by  Formula  ( l ). 

1.  Factor    a'  +  2ah  -f  ly"  A^is.    [a  +  5)  (a  -f-  b), 

2.  Factor    ia?  +  Vlab  -f  9Z>2. 

^  Aois.   {2a  +  31)  (2a  +  3^). 

3.  Factor    9a^  +  I2ab  +  4R 

A91S,   (Sa  +  2b)  {3a  +  2b). 

4.  Factor    4x^  -\-  8x  -^  4.  Ans.    {2x  -f  2)  {2x  -f  2). 

5.  Factor    9^252  +  l^abc  +  4c2. 

A71S.    {Sab  +  2c)  {dab  -\-  2c). 

6.  Factor    IQx^y^^  +  IQxy^  +  4y\ 

Ans.    {Axy  +  2y'^)  {4xy  +  2y^). 

63.  When  two  terms  of  a  trinomial  are  squares,  and 
positive,  and  the  third  term  is  equal  to  minus  twice  their 
square  roots,  the  trinomial  may  be  factored  by  Formula 
(2). 

1.  Factor  a^  _  2ab  +  b\  Aiis.    {a  -  b)  {a  -  b). 

2.  Factor  An?;  —  4ab  +  R  Ans.    {2a  —  b)  {2a  —  b). 

3.  Factor  Oa^  _  Qac  +  €\  Ans.    {3a  —  c)  {3a  —  c). 

4.  Factor  a'^x'^  —  4ax  -f  4.  A7is.    {ax  —  2)  {ax  —  2) 

5.  Factor  4x^  —  4:cy  -f  y"^-  Ans.    {2x  —  y)  {2x  —  y) 

62.  Wbi'n  may  a  trinomial  be  factored  ? 

68.  When  may  a  trinomial  be  factored  by  this  method? 


KAOTORINO.  81 

6,  Factor    ZQx^  —  24ay  -f  4y2. 

.1;^.^.    (Gx  —  2.v)  (6a;  -  2y). 

64.  Wlien  the  twd  terms  of  a  l/momial  are  squares  and 
have  contrary  signs,  the  biuoniial  may  l)e  factored  by 
FoiTiiula  ( 3  ). 

1.  Factor    4c2  —  b'i  Ans,   (2c  -f  b)  (2c  —  b) 

2.  Factor    8la^c^  -  db'c^. 

A?is.   (0(fc  +  Uc)  (Oac  —  Sbc). 

8.  Factor  64<z'b^  —  IhxHf. 

Ans.   (Sa'^b'^  -f  Bxij)  (Sa'^b^  -  5xi/). 

4.  Factor    25a'^c'^  -  9x*;/\ 

Afis.    (oac  +  Sx^i/)  (one  —  3u*2y). 

6.  Factor    3Ga*b*c^  -  Ox". 

Ans,   (Qa^b'^c  +  Ba^)  (Ga^b^c  -  iix-^). 

r. 

0.  Factor    49aj*  -  3Gy*.     ^/w».    (1x^  +  6y2)  (Ta-  —  ny*'). 

e.'i.  When  the  two  terms  of  a  Vinomial  are  eulx  s,  niul 
have  ex>ntrary  signs,  the  hinotnial  may  be  fao'o'.oil  l.y 
Formula  ( 4  ). 

1.  Factor    %d?  —  c^.        Ans.  (2a  —  c)  (\a'^  a-  e^), 

2.  Factor    2nd?  —  64. 

Ans.    (3a  -  4)  (Qa^  4    1  _  /  \-  16). 

3.  Factor    a?  —  G^h\ 

Ans.    (a  —  4b)  (a^  -f  iab  -f  ^Gb^), 

4.  Factor    a^  —  27R     ^??.9.    (a  -  3^»)  (a'  +  3or^  +  Oi'O- 

r»4.  Wlien  may  a  binomial  be  factored  ? 
<\6.  When  mnr  a  binomml  bo  fvjior<'<l  »)v  \h  i»  inothoril' 
4* 


82  ELEMENTARY       ALGKRRA. 


(/« 


66.  When  tlie  terms  of  a  binoiiiial  are  cubes  and  have 
ike  signs,  the  binomial  may  be  factored  by  F'ormula  (  5  ). 

1.  PVctor    Sa^  +  cK      Ans,    (2a  -f-  c)  (4a2  _  2ac  +  c'). 

2..  Factor    21  a^  -^  64. 

Ans.    (3a  -j-  4)  (Oa^  -  \oa   f  10). 
6:   Factor    a^  -|-  34^>3. 

^??s.    (a  -f  Ab)  [a^  -  iab  4-  166*^). 

4,  Factor    a^  f  21  b\     A?}s.   (a  -f-  3*)  (a^  _  8or^>  -f  9^>^). 

67.  When  the  terms  of  a  binomial  are  4th  powers,  and 
have    contrary    signs,    the    binomial    may    be    factored    by 

I     Fonnula  (6). 

1  What  are  the  factors  of  a"  —  b^? 

Ans.   (a   i-  b)  {a  -  b)  [rf  -f-  -6^). 

2  Wliat  are  the  factors  of    81a*  -  T6^;^  ? 

Ans.    (3a  +  2h)  (3a  -  2b)  (Oa*^  +  4*^). 

3.    What  are  the  factors  of    16a-^/^^  —  Slc^c/*  ? 

Ans.    {2(tb  ^  3c(0  (2^/^  —  3cc/)  {ia'^b'^  +  9e''c/2)^ 


GUEATEST     COMMON     DTTTSOR. 

68.  A  Common  P«tvtsok  ot  two  quantities,  is  a  quantity 
that  will  divide  them  both  wit^hout  a  remainder.  Thus, 
ba-'^'J^is  a  common  divisor  of  da'^b^c  and  3aV/'*  —  Qa^P, 

fi^  When  may  a  binomial  be  factored  by  this  method? 
f>7.  When  may  a  binomial  be  factored  by  this  method? 
68.    What  is  the  oominon  divisor  of  two  anantitieg  )r 


GREATEST      COMMON       DIVISOR.  83 

69.  A  Simple  or  Prime  Factor  is  one  that  cannot  be 
resolved  into  any  other  factors. 

Every  prime  factor,  common  to  two  quantities,  is  a  com- 
mon flivisor  of  those  quantities.  Tlie  continued  product  of 
any  number  of  piime  factors,  common  to  two  quantities,  is 
also  a  common  divisor  of  those  quantities. 

TO.  The  Greatest  CoarMON  Divisor  of  two  quantities, 
IB  the  continued  product  of  all  the  piime  factors  which  are 
common  to  both. 

71.  When  both  quantities  can  be  resolved  uitb  prinm 
fiictors,  by  the  method  of  factoiing  already  given,  the  great- 
est conmion  divisor  may  be  found  by  the  following 

rule. 

L  Resolve  both  quantities  into  their  prime  factors  : 
II.  Fiiid  tlie  continued  product  of  all  the  factors  which 
are  common  to  both  /  it  will  be  the  greatest  common  dv)i- 
9or  required. 

EXAMPLES. 

1.  Required  the  greatest  common  dinsor  of  I5a^^c  and 
25abd.     Factoring,  we  have, 

Iba^b'^c  =  3  X  5  X  b'aabbc 
2^nhd  =  5  X  babd. 

Tlie  factors,  5,  5,  a  and  i,  are  common ;  hence, 

5x5Xax6  =  25aft, 

IS  the  divisor  sought. 

69.  What  is  a  simple  or  prime  factor?     Is  a  prime  factor,  common  to 
two  quantities,  a  common  divisor? 

70.  What  is  the  grcHtcst  common  divisor? 

71.  If  both  quantities  can  be  renolved  into  prime  fnctore  how  do  you 
6ud  the  ijroatefit  »onimoii  divipory 


84  EIEMENTARY       ALGEBRA. 


VERIFICATION. 

nha^lP-c  -^  Toah  -  Zahc 
Ihabd  -V-  25aZ>  =  d\ 

and  since  the  quotients  lia\e  no  common  factor,  they  cannot 
be  further  di\dded. 

2.  Required  the  greatest  common  divisor  of  a?  —  2ab  4 
h"^  and  a?  —  IP-.  Ans.  a  -   b 

3.  Required  the  greatest  common  divisor  of  a^  +  2ab  -\ 
b^  and  a  -i-  b.  A7is.  a  -\-  b 

4.  Required  the  greatest  common  divisor  of  a^x^  —  4ax 
4-  4  and  ax  —  2.  Ans.  ax  —  2. 

5.  Find  the  greatest  common  divisor  of  3a^&  —  Qa'^o 
•^  1  %a^xy  and  b'^c  —  Sbc'^  —  Qbcxy.     Ans.  b  —  Sc  —  Q>xy. 

6.  Find  the  greatest  common  divisor  of  Aa^c  —  iacx  and 
iia^g  —  Zagx.  Ans.  a{a  —  x)^  or  a^  —  ax. 

1.  Find  the  greatest  common  divisor  of  4c'^  —  \2cx  +  Occ^ 
and  4c2  —  Qx^.  Ans.  2c  —  dx. 

8.  Find  the  greatest  common  divisor  of  x^  —  y^  and 
a.2  _  y2^  Ans.  X  —  y. 

9.  Find  the  greatest  common  divisor  of  Ac^  -f  45c  -f  b"^ 
and  4c2  —  6^.  ^It?^.  2c  -f-  5. 

10.  Find  the  greatest  common  divisor  of  2ba\Z.—  9x^y* 
and  r^acd^  +  Sd^x^Y-  ^^^'  -«^^  -^-  3a;V. 

NoTE.-^To  find  the  greatest  con^mon  divisor  of  three 
quantities.  First  find  the  greatest  common  divisor  of  two 
of  tliem,  and  then  the  greatest  common  divisor  betv>^een  this 
result  and  the  third. 

1.  What  is  the  greatest  common  divisor  of  4ax'^y,  lQabx% 
and24aca.'^?  Ans.  ^ax^, 

2.  Of  3£c2—  6fc,  1x^—  4x\  and  cc^ /_  ^xyl    Ans.  x"'—  2iC 

•72.  When  is  one  quauilty  a  tntiltiple  of  another? 


LEAST      COMMON      MULTIPLE.  85 


LEAST    COMMON     MULTIPLE. 

79,  One  quantity  is  a  mlt>tiple  of  another,  when  it  can 
be  divided  by  that  other  without  a  remainder.  Thus,  Bd^d, 
U  a  multiple  of  8,  also  of  a%  and  of  b.  * 

73.  A  quantity  is  a  Common  Multiple  of  two  or  moro 
quantities,  when  it  can  be  divided  by  each,  separately,  witli- 
out  a  remainder.  Thus,  24a^ar',  is  a  common  multiple  of 
Goaj  and  4a^£c. 

74.  The  Least  Common  Multiple  of  two  or  more  quan- 
tities, is  the  simplest  quantity  tliat  can  be  divided  by  each, 
without  a  remainder.  Thus,  12 a^b^x^,  is  the  least  common 
multiple  of  2d^,  4ab\  and  6a^b^^. 

75.  Since  the  common  multiple  is  a  dividend  of  each  of 
the  quantities,  and  since  the  division  is  exact,  tlie  common 
multiple  must  contain  every  prime  factor  in  all  the  quanti- 
ties ;  and  if  the  same  factor  enters  more  than  once,  it  must 
enter  an  equal  number  of  times  into  the  common  multiple. 

When  the  given  quantities  can  be  factored,  by  any  of  the 
methods  already  given,  the  least  common  multiple  may  be 
found  by  the  following 

BULB. 

I.   JResolve  each  of  the  qiicmtities  into  its  prime  factors : 
IT.    Take  each  factor  as  many  times  as  it  enters  any  07ie 
of  the  quantities^  and  form  the  continued  produi't  of  these 
factors  /  it  will  be  the  least  common  midtiple. 

78.  When  is  a  quantity  a  common  multiple  of  several  others? 
74.  What  is  the  lcj\st  common  multiple  of  two  or  m6re  quantitins? 

76.  What  does  the  common  miiltiple  of  two  or  more  qvianiities  contain, 
■IS  factors?    How  may  the  least  common  multiple  be  found  ? 

*  Th«  mulUpU  <A  «  qnftotltj,  is  Rtropl}  %  dimkUnd  whicb  will  glv«  an  exset  quotient 


S6  E  L  E  M  E  N  T  A  K  Y       ALGEBRA. 

EXAMPLES. 

1.  Find  the  least  common  multiple  of  X^a^li^c^  and  8«^fr\ 

Sa'^b'^  r-   2.2.2.<mbbb. 

Now,  since  2  enters  3  times  as  a  factor,  it  must  enter  3 
times  in  the  common  multiple :  3  must  enter  once ;  a,  3 
times ;  b,  3  times ;  and  c,  twice ;  hence, 

2.2.2.3acfabbbcc  =  24:a^b^c\ 

is  the  least  common  multiple. 

Find  the  least  common  multiples  of  the  following : 

2.  6a,  5a%  and  25abc'^.  Ans.  IBOa^bc^ 

3.  Sa^b,  9abc^  and  2la?x^.  Ans.  21d^bcx^, 

4.  4«2ic2y2,  Sa^xy,  16a*y^,  and  24a^y'^£c.     Ans.  iSa^x^y*, 
b.  ax  —  bx^  ay  —  by^  and  x'^y'^. 

Alls,  {a  —  b)x.x.yy  —  ax^y"^  —  bx-y'^. 
6.  a  -^  b,  a?  —  ^^  and  a?  +  2a^>  +  b"^. 

Ans.  {a  +  ^')2  (a  -  b), 
1.  Sa^b\   Qa^x\  18«y,  Sa^y^.     .  Ans.  IStrb^x'^yl 

8.  Qa%a  i  b),  I5a\a  -  by,  and  1 2^3(^2  _  ^2). 

.  A/is.    \20a^{a  —  b)'^  {a  +  S). 


FRACTIONS.  87 


CILVPTER  rV. 

FRACTIONS 


76      If  the  iinit  1  be  divided  into  any  number  of  eqnal 

parts,  tfacli  pail  is  called  a  FRAcnoNAL  umt.    Thus,  -  ,  j, 
11  2     4 


^  ,  y  ,    are  fractional  units. 


77.     A  Fraction  is  a  fractional  unit,  or  a  collection  of 
fractional  units.    Thus,  - ,  - ,  -  ,  ^ ,  are  fractions. 


7§.  Every  fraction  is  composed  of  two  parts,  the  De- 
nominator and  Numerator.  The  Denominator  sliows  into 
how  many  equal  parts  the  unit  1  is  divnded ;  and  the  Nur 
inerator  how  many  of  tliese  parts  are  taken.     Thus,  in  the 

fraction    i ,  the  denominator  J,  shows  that  1  is  divided  hito 

0 

h  equal  parts,  and  the  numerator  a,  shows  that  a  of  these 
parts  are  taken.  The  fractional  unit,  in  aU  cases,  is  equal  to 
the  reciprocal  of  the  denominator. 

7fi.  If  1  be  divided  into  any  number  of  equal  parts,  what  is  each  part 
called? 

77.  What  is  a  fmction  ? 

78.  Of  how  many  parts  is  any  fraction  composed?  What  arc  they 
oatled?  What  doesi  the  denominator  show?  What  the  numerator? 
VVhrii  ie  r.h»»  frn<'tioii:il  iiiiif  eiju.il  to? 


88  E  L  E  :M  E  N  T  A  R  Y       ALGEBRA. 

70.  An  Entire  Quantity  is  one  which  contains  no 
fractional  part.  Thus,  7,  11,  a^x^  ix^-  —  3y,  are  entire 
quantities. 

An  entire  quantity  may  be  regarded  as  a  fraction  whose 

denominator  is  1.     Thus,  7   =   - ,    a5  r=  —  • 

80      A  Mixed  Quantity  is  a  quantity  containing  both 

bx 

entire  and  fractional  parts.    Thus,    7p*o>  ^^  »  ^  H >    ^^^ 

c 

mixed  quantities. 

SI.     Lf^t  7    denote   any   fraction,   and    q   any  quantity 

.  d 

whatever.     From  the  preceding  definitions,   -  denotes  that 

7    is  taken  a  times ;  also,    -r^    denotes  that    y    is  taken 

0  0  0 

aq  times ;  that  is, 

aq         a  , 

-~  —  -   X   q\  hence, 

Multiplying  the  numerator  of  a  fraction  by  any  quan- 
tity^ is  eqxdvalent  to  rnultiplymg  the  fraction  by  that 
quantity. 

We  see,  also,  that  any  quantity  may  be  multiplied  by  a 
fraction^  by  multiplying  it  by  the  numerator^  and  then 
dividing  the  result  by  the  denonfiinator. 

82.  It  is  a  principle  of  Division,  that  the  same  result  will 
be  obtained  if  we  divide  the  quantity  a  by  the  product 
of  two  factors,  ji?  x  5',  as  would  be  obtained  by  dividing  it 

79.  What  is  an  entire  quantity  ?  "When  may  it  be  regf.rded  as  a  frao 
lion  ? 

SO.  What  is  a  mixed  quantity  ? 

81.  How  may  a  fraction  be  rauUipliod  by  any  quantity  ? 

82    How  may  a  traction  be  divided  by  any  quantity  ? 


TBANSFOKMATION      OF      FRACTIONS.  89 

first  by  one  of  the  fiictors,  jt),  and  then  dividing  that  result 
by  the  other  factor,  q.    That  is, 

—  =  (-)-f-^;    or,    —  z=  \-\  -^  p\  hence. 

Multiplying  the  denominator  of  a  fraction  by  any  quan 
iity^  is  equivalent  to  dividing  the  fraction  by  that  quantity 

83.  Since  the  operations  of  Miilti]>lication  and  Division 
are  the  converse  of  each  other,  it  follows,  from  the  preced 
ing  principles,  that, 

Dividi7ig  the  numerator  of  a  fraction  by  any  quantity^ 
is  equivalent  to  dividing  the  fraction  by  that  quantity ; 
and. 

Dividing  the  denominator  of  a  fraction  by  any  quantity ^ 
is  equivalent  to  multiplying  the  fraction  by  t/uit  qtiantlty, 

84.  Since  a  quantity  may  be  multiplied,  and  the  result 
divided  by  the  same  quantity,  without  alteiing  the  value, 
it  follows  that, 

Both  terms  of  a  fraction  may  be  multiplied  by  any  quaty- 
tity,  or  both  divided  by  any  quantity^  without  changing  the 
value  of  the  fraction. 


TKANSFORMATION     OF     FRACTIONS. 

85.  The  transformation  of  a  quantity,  is  the  operation 
of  changing  its  form,  without  altering  its  value.  The  tenn 
reduce  has  a  technical  signification,  and  means,  to  IVaiw- 
form, 

98.  What  follows  from  the  preceding  principles  ? 
84.  What  operations  may  be  performed  without  alf^ring  tho  value  of 
a  fraction  ? 

86.  What  is  the  tranpformation  of  a  quantity  ? 


90  ELEMENTARY       ALGEBRA 


FIKST  TRANSFORMATION. 

To  reduce  an  entire  quantity  to  a  fractional  form  having  a 
given  denominator. 

86.  Let  a  be  the  quantity,  and  h  the  given  denomi- 
nator.   We  have,  evidently,    a  =  — ;    hence,  the 

RULE. 

Multiply  the  quantity  by  the  given  denominator,  and 
write  the  product  over  this  given  denominator, 

SECOND   TRANSFORMATION. 

To  reduce  a  fraction  to  its  lowest  terms. 

87.  A  fraction  is  in  its  lowest  terms^  when  the  numerator 
and  denominator  contain  no  common  factors. 

It  lias  been  shoAvn,  that  both  terms  of  a  fraction  may  be 
divided  by  the  same  quantity,  without  altering  its  value. 
Hence,  if  they  have  any  common  factors,  we  may  strilie 
them  out. 

RULE. 

Resolve  each  term  of  the  fraction  i?ito  its  prime  fac- 
tors /  then  strike  out  all  that  are  common  to  both. 

The  same  result  is  attained  by  dividing  both  terms  of  the 
fraction  by  any  quantity  that  wilJ  divide  them,  without  a 
remainder ;  or,  by  dividing  them  by  their  greatest  common 
divisor. 


86.  How  do  you  reduce  an  entire  quantity  to  a  fractional  form  having 
a  given  denominator  ? 

87.  How  do  you  reduce  a  fraction  to  its  lowest  terms  ? 


TRANSFORMATION      OF      FRACTIONS.  91 


EXAMPLES. 

I.  Redace    — — -,  to  its  lowest  terms. 

Canceling  the  common  factors,  5,  a,  and  c^  we  have, 
25ac5  —  1J 


A71S, 


2.  Reduce 

8.  Ro<luce 

.  4.  Reduce 

^    6.  Reduce 

,  6.  Reduce 

• — 7.  Reduce 

8.  Reduce 


ab  —  oc 
b  -  c  ^ 
y^2  _  2w  4-  1 
^-  1 
g3  -  ax'i 

4Qa*b^  —  66a^  *  **'""   8a*  —  lla'^^»^ 


^n«. 

17 

^Irw. 

5c 

Ans. 

a 

1 

=  a 

Ans. 

n 

-  1 
+  1 

Ans, 

X 

a;2 
—  a 

-4/w. 

8 

1 

= 

-8. 

rlTia. 

4ft  - 

-  ( 

3a 

• 

a2  -  62  a  +  5 

9.  Reduce    -r -,.,,'  ^w«.  ^ r- 

a*  —  2ab  +  P  a  —  b 

J    10.  Reduce    -^ --4r ^^"f-   -^-5 ^ 

8a3  —  8a'6  8a 

3a2  +  6a2ft2  .           1  +  2ft2 

11.  Reduce    ,^„,  .    .^,..,'  ^««. 


12a*  +  6a V  4a2  -|-  2a6"* 

12.  Reduce    — ry-^ — ::i\  — •  ^^'    0/ " 

3(a2  —  a;«)  3(/?  —  a;^ 


WZ  ELEMENTARY       ALGEBRA.. 

THIRD   TRAXSPORilATIOX. 

To  reduce  a  fraction  to  a  mixed  quantity, 

88.  When  any  term  of  tlie  numerator  is  divisible  by  any 
term  of  the  denominator,  the  transformation  can  be  effected 
by  Dixision. 

RULE. 

Perform  the  indicated  division^  continuiyig  the  operation 
as  far  as  possible  ;  then  write  the  remainder  over  the  deno- 
minator^ and  annex  the  result  to  the  quotie^it  found. 


^    „    .  ax  —  a 

1.  Reduce 


2.  Reduce 

3.  Reduce 


EXAMPLES, 

2 


X 

ax 

— 

a;2 

X 

ah 

— 

2a2 

h 

«2 

— 

^2 

Ans.  ( 

2  - 

X  ' 

Ans. 

a 

—   X. 

Ans,   a 

— 

Ans. 

a 

:i 

(  x"  -  ^'3  ■     »- 


5.  Reduce'   — —-    j  Ans.   x^  -f  xy  -}-  ij^, 

%   X  —  y       -    ' 

6.  Reduce Ans.   2a;  —  1  +  ---- 

bx  OX 

^    „    ^  36^3  -  72aj  4-  32a2a;2  32^/^2; 

7.  Reduce ^ . .  ix^  —  Q  -\-  —-  - 

9a;  9 

,,    ,  \Sacf  —  (ihdcf  —  2ad  '         6c       2hc        2 

8.  Reduce .y •  •    ^- t:-. 

',U(df  d         a        3/ 

^    ,,    ,  a!2  +  a;  -  4  .  ,  2 

9.  1  deduce •  Ans.    ic  —   1 

a;  +  2  at  +  2 

68.  How  do  you  reduce  a  fmction  to  a  mixed  quij  tJt,y  f 


TRANSFORMATION       OF      FRACTION  t?.  iK. 

d^  _L  1)1  26* 

10.  Reduce  — -—r  •  Am.  a  —  h  -\- 


a  ^  h  '  a  4-6 

11.  Reduce  ?L±i^J5.  A^is.  x  +  1  -{-  — 5— 


-i 


FOURTH    TRANSFORMATION. 

7b  reduce  a  mixed  guant.it i/  to  a  fractional  form. 

89.  This  transformation  is  ilu'  converse  of  the  preced- 
ing, and  may  be  effected  by  the  following 

BU  LE. 

Multiply  th>e  entire  part  by  the  denomirmtor  of  the  frac- 
tion^ and  add  to  the  product  the  mmierator  ;  write  the  resuU 
over  the  denominator  of  the  fraction, 

EXAMPLES. 

1.  Rednce  6^  to  the  form  of  a  fraction. 

JO 

6  X  V  =  42 ;   42  +  1   =  43 ;  hence,  6^   =.  y  • 
Reduce  the  following  to  fractional  forms : 


2. 

X  —^ ^  = 

X 

-a^) 

2a?  -  a» 
Ans. < 

X 

3. 

ax  •+  Q^) 
la      • 

.       ax  —  Q? 

Ans.  — 

2a 

4. 

-^- 

Am,  — 

3a; 

5. 

^       JB  -  a  -  1  ^ 
a 

.        2a  —  a;  -t-  1 

Am, ^— 

a 

6. 

5a! 

A 

ns. 

10.r2  4-  4ar  +  3 
hx 

80.  How  do  you  reducp  a  mixed  quantity  to  a  fractional  form? 


94  ELEMENTARY        ALGEBRA 


H     ^      .    r        3c  4-  4  ^        16a  -\-  Sh  --  Sc  -  4 

7.  2a  +  0 —  •  A?is. 

8  8 

^     ^        .    ,        Ga^a;  —  ad  .        18a'^x  +  5ab 

8.  6aic  +  6 Ans. 


9.    8  +  Sab  — 


4a  4a 

8  +  Qa'^b'^x\ 
I2abx* 

9Qabx^  +  30^2520;*  —  8 


A?is. 


I2abx^ 


FIFTH    TRANSFORAtATION. 


To  reduce  fractions  having  different  denominators^  to  equi- 
valent fractions  having  the  least  common  denominator. 

90.  This  transformation  is  effected  by  finding  the  least 
common  multiple  of  the  denominators. 

-10  c 

1.  Reduce  -,  -,  and  — ,  to  their  least  common  denomi- 
nators. 

The  least  common  multiple  of  the  denominators  is  12, 
which  is  also  the  least  common  denominator  of  the  required 
fractions.  If  each  fraction  be  multiphed  by  1 2,  and  the  resiill 
divided  by  12,  the  values  of  the  fractions  will  not  be  changed, 

-  X  12   =  4,     1st  new  numerator ; 

g 

-  Xl2   =  9,     2d   new  numerator : 
4 

—  X  12   =  6,     3rd  new  numerator ;  hence, 

4      9^5  ^  .     ,        ^       . 

-— ,  -— ,  and   —  are  the  new  equivalent  tractions. 

\  i  \  2d  J.  Z 

00.  How  (10  you  reduce  fractions  having  different  denominators,  to  equi 
\  alent  fractions  having  the  least  common  denominator  ?  When  the  nu- 
merators have  no  common  factor,  how  do  you  i  educe  tiiem  ? 


TRAHSFOKMATION   UF   FKACTIONS.   95 


BULB. 

L   J^nid  the  least  common  multiple  of  the  denomwatora : 
II.    Multiply  each  fraction  by  it.,  and  cancel  the  denon^ 

inaioT  : 
m.    Write  each  product  over  the  common  multiple,  and 

tjuj  res'ults  will  be  the  required  fractions, 

GENE  HAL      BULE. 

Multiphj  each,  numeraior  by  all  the  denominators  except 
its  own,  for  t/ie  new  numerators,  and  all  the  denominati/rs 
togetfierfor  a  common  denominator. 

EXAMPLES. 

a  c 

1.  Reduce  -r rs  and  — ; — =   to  their  least  common 

a}  —  b^  a  -\-  b 

denominator. 

The  least  common  multiple  of  the  denominators  is  (a  +  J) 
('/-ft): 

^-^2  X  (a  +  5)  («  -  5)  =  a 

— — T  X  {a  -\-  b)  {a  —  b)  =  c{a  —  b;  hence, 

and    —z\/      — T\y   are  the  required 


(a  +  b)  {a  —  b)  {a  +  b)  (a  -  b) 

fractions. 

Reduce  the  following  to  their  least  common  denominators ; 
^    3a;      4  ^    12«2  45a.    40     48a;» 

^'  4'  6»  '""^   ■T5--  ^^^'-  -eo'e-o'lcr 

8.C/,   --,    and    -5-.  Ar^.   — ,  -,  -- 

^    Zx      2b  ",  .         Of-a:     4ab     Qacd 

4.  —-  ,    —  ,    and    d.  Ans.    - —  ,  - —  ,  — — 

2a'    3c  6ac     6ac      Que 


96  E  L  E  M  E  N  T  A  li  Y       A  L  G  K  B  R  A  . 

(X. 

3     2x  2x  .  9a      Sax     I2a^  -\-  2ix 

'•  4'  ¥'  «  +    a  •  ^'''-    12^'  m'  ^^m"^' 


a 


.r-2 


6.   ' — : ,  -— rr, ,  and 


1  -  x\{l  -  xy^'  (1  -  xy- 

'  x(l  —  xY    a;2(i  -  x)         ,         x^ 

.      c      c  —  b         ,       c 

7.  -^  ,  ,  and  — -—=■ ' 

ac^  +  bc^       5a^c  —  5a^b  +  5abc  —  5ab^  5ac^ 

5tt-c  +  5abd'  5a^c  -\-  babe  '   ba^c  +  babe ' 

ex  dx^  '       x^ 

o. ,  — - —  ,  and 


a  —  a^'aH-a:'  a  -\-  x 

.         cxia-\-x)     dx^ia—x)        \  x^ia  —  x) 

Arts.    -\--^  ,  — ^ ^  ,  and  -\ ^ 

a^  —  x^        a^  —  x^  a^  —  x*- 


ADDITION    OF    FEACTIOISTS. 

91.  Fractions  can  only  be  added  when  tliey  have  a  com- 
mon unit,  that  is,  when  they  have  a  common  denominator. 
In  that  case,  tlie  sum  of  the  numerators  av411  indicate  how 
many  times  that  unit  is  taken  in  the  entire  collection. 
Hence,  the 

RULE. 

I.  Reduce  the  fractions  to  be  added,  to  a  common  denom^ 
inator  : 

n.  Add  the  numerators  together  for  a  new  numerator^ 
and  write  the  sum  over  the  common  denominator, 

EXAMPLES. 
6      4  2 

1.  Add    -,  -,  and  -,  together. 
2     3  5 

91.  Whftt  ip  the  rule  for  adding  fractions  ? 


ADDITION      OF      FRACTIONS 


97 


By  reducing  to  a  common  denominator,  we  have, 

6  X  3  X  5   =  90,  Ist  numerator. 

4  X  2  X  6  =  40,  2d  numerator. 

2x3x2  =  1 2,  3d  numerator.        » 

2  X  3  X  6  =  30,  the  denominator. 

Hence,  the  expression  for  the  sum  of  the  fractions  becomes 

90       40       12  _  142^ 
80       30  "^  30  "~    30  *' 

which,  being  reduced  to  the  simplest  form,  gives  A\^. 

(Z      C  B 

2.  Find  the  sum  of  y>  -t>  and  -• 
0    a  f 

Here,     a  x  d  x  f  —  acif   ) 
c  X  b  X  f  =   cbf 
e  X  b  X  d  =   ebd 


bnd 


b  X  d  xf  =  bdf 


„  adf  ,    cbf  ,   ebd        adf  +  cbf  +  ebd     . 


the  new  numerators. 

the  conMnon  denominator, 
sum. 


Add  the  following : 

3aj2         _  ^    ,   2flKB 

3.  a =- ,  and  b  + 

o  c 


Ans.  a  +  b  + 


2abx  —  3ca^ 


X        X 

2'    3' 

SB 


and 


-2        -    4a; 
- —   and    r-» 
3  7 


6.  a;  H —  and    3a;  -\ — 

^    .       ^^         ^    «  +  « 
^-  -*«,    — ,    and 


Ans.  X  + 
Ans. 
Ans.  Ax  + 


be 

X 

12 
19a;  —  14 

21 
10a;  —  17 


8. 


2a; 


2a 

1x 


3'    T 


and 


2a; 
2x  4-  1 


Ans.  Ax  4- 


Ans,  2x  -h 


12 

5x^  +  ax  -^  a^ 
2nx 
A9x  -I-  12 


60 


9.  4(8,    ~,    and    2  -}-  | 


Ans.   2  4-  4a  4- 


44a: 
'45' 


98  'elementary     algebra. 

10.  3a;  +  ^   and   a;  -  ^,  Ans.   3a;  +  — 

5  9  '45 


11.  ac  —  --   and    1 -,- 

8a  <^ 


Ane,   1  +  ac  — 


>^ 


8at? 

q  q  J 

^715. -— 

{X   -    1)3 

13.    rr— - — r,  -7- r,  and  777- ^.  •    -4?15. 


4(1  +  a)'  4(1  -  a) '       "  2(1  ^-  a^)  I  -  a^ 


SUBTEACTION    OF    FKACTION'S. 

92.  Fractions  can  only  be  subtracted  when  they  have 
the  same  unit;  that  is,  a  common  denominator.  In  that 
case,  the  numerator  of  the  minuend,  miiius  that  of  the  sub- 
trahend, will  indicate  the  number  of  times  that  the  common 
unit  is  to  be  taken  in  the  difference.     Hence,  the 

KULE. 

I.  Reduce    the  two  fractions  to  a  common   denomi- 

inator  : 

n.  Theii  subtract  the  numerator  of  the  subtrahend  from 
that  of  the  minuend  for  a  new  numerator^  and  write  the 
remainder  over  the  common  denominator. 

EXAMPLES. 

3  2 

1.  VThat  is  the  difference  between  -  and  -  - 

7  8 

3        2_24_U_10_^        . 
7  "~  8   ~   56  ~  56   ~  56  ""   28 '       ^^' 

92.  What  is  the  rule  for  subtracting  fractions 


MULTIPLICATION      OF      FEA0TI0N6.  99 

ic  —  cb  2<2  —  4cc 

2.  Find  the  diiference  of  the  fractions  -—rr—  and  — 

26  3c 

TT  S      (a;  —  a)  X  3c  =  3caj  —  3ac  )    . 

Here,    1  ,„^        ,  (       „,         ,    ,       „.    Khe  numerator?, 
'    {{2a  —  4a;)  x  26  =  4a6  —  86a;  ) 

and,  26  X  3c  =  G6c  the  common  denominator. 

_-  Sob— Sac      4a6— 86a;       3ca;— 3«c— 4a6-{-86a;     . 

Hence,  — --. -r =  —: — -—^ Aiis, 

'      66c  66c  66c 

3.  Required  the  difference  of  -r--  and  — -  •       Ans,  —^  • 

^  '7  5  35 

4.  Required  the  difference  of  6y  and  -^  •        Ans.  -~  • 

6.  Required  the  difference  of  —  and  —  •  Ans,  -—  • 

(J.  From  't±JL  subtract  ^-=^  •  Ans.  ■  4^,  • 

a;  —  y  x -{-  y  x^  —  yK 

7.  From    \'"      subtract  -r -•         Ans.  ^ — r-  • 

y  —  z  y^  —  z^  y^  —  '^ 

Find  the  differences  of  the  folloTsing : 

^    3a;  +  a       ,  2a;  4- 7        .       24a;  +  8a  —  106a;  —  356 

8.  — -£ —  and  — —— •    Ans,  rr^ 

56  8  406 

^«      .a;       -  a;  —  a      .       ^     ,    ex  -^  bx  --  ab 

9.  3a;  +  J  and  x Ans.  2x  -{ z » 

p  c  oc 

-^■^        ,a  —  a;         ,a  +  a;  .  4a; 

10,     a  H — 7 — ; — r  and  —, :•      Ans,  a :; :^- 

a(a  +  X)  a{a  —  a;)  a^  —  ar 


MUlfflPLIOATION    OF    FRACTIONa 

CL  G 

93»    Let  T  and  -^  represent  any  two  fractions.    It  hsjt\ 
been  sliown  (Art.  81),  that  any  quantity  may  be  multiplied 

98.  What  is  the  rule  for  the  multiplication  of  fractions  f 


iOO  ELEMENTARY      ALGEBRA. 

by  a  fraction,  by  first  multiplying  by  the  numerator,  and 
then  dividing  the  result  by  the  denommator. 

To  multiply  j-  by  -  ,  we  first  multiply  by  c,  givmg  ~ 
then,  we  di\^de  this  result  by  c7,  which  is  done  by  multiply- 

CLG 

ing  the  denominator  by  d ;  this  gives  for  the  product,  j-^ ; 

that  is, 

a       c         aQ      , 

RULE. 

I.  If  there  are  mixed  quantities^  reduce  them  to  a  frao 
lional  form  ;  then^ 

n.  Multiply  the  numerators  together  for  a  new  numera- 
tor^ and  the  denominators  for  a  new  denominator. 

EXAMPLES. 

1.  Multiply  a  -\ hy  -•     First,   a  -\ =  , 

a  a  a  a 

a"^  +  hx       c        a^c  •\-  hex        . 

hence, X  -,  =  ^ —  •     Ans. 

^  a  d  ad 

Find  the  products  of  the  following  quantities : 

^    2a;     3a5         ,  Zac  . 

2.  — ,  ,  and  — =-  Ans.  9ax, 

a'     c   ^  2b 

^    ■.    ,   hx       ^  a  .        ah  -\-  hx 

3.  0  H and  -•  Ans. • 

ax  X 

,    x^  — J2       ^  a;2  +  52  .  a;*  —  5* 

4.  — r —  and  -^ •  Ans. 


be               b  +  c  '  bH  -\-  bc^ 

aj-fl         ^  X  —  \  .        ojx^  —  ax  4-  x^  —1 

6.  a  H ,  and  — — r*     Ans. -r— — -, ■ 

a     '          a  -^  b  a^  -\-  ah 

■      ■    ,       ax  *        ,  a^  —  a;2  .        a^  4-  «^ 

6.  a   j ana  — ; — r--  Ans. ; — ^ 

a  —  X          x  -\-  x^  x  -\-  x^ 


MULTIPLICATION      OF      FBAOTIONB.  101 

7    Multiply    7    by • 

Of    —    U  O 

We  have,  by  the  rule, 

2a  g}-  b^  __  2a(a^-,  &^)    .  .  .^(^a.  ^  .b\  (a  -  b) 

a  -  b  ^        3        ~~     S{a'—  b)     "^i '  '■ . •'  >^aV; f/). 

=  f  (a  +  i). 

After  indicating  the  operation,  we  factored  both  numera* 
tor  and  denominator,  and  then  canceled  the  common  factors, 
before  performing  the  multiplication.  This  should  be  done., 
w/ienever  there  are  common  factors. 


8. 

2        .     x^-y' 

^m,  2(^  +  2'' 

X  —  y~       *'          a 

a 

9. 

a;2  _  4                    Ax 
3             ^      a;  +  2 

A^.  "<'-». 

10. 

{a^by                   Ax^ 
2x           ^       {a  +  b) 

Arts,   2x{a  +  b\ 

11. 

y3              ^          x-l 

y 

12. 

1  -  a;2          ^      a  -{-  X 

4         a  —  X 
Arts. • 

1    —  X 

13. 

x^     2.y                 ^         2xy 
x-y                       x-Yy 

Arts.  a^. 

14. 

2a  -  b               6a  —  25 
4a          ^^      62  _  2a& 

.        i  -  3a 
Ans,      ^   .     '     ^ 
2ab        .'^ 

14. 

x-?^'     by      ?f2/. 
a:        "^      y       a 

Ans,  — — ^ 

^y 

102  ELEMENTARY      ALGEBRA, 


DIYISIOlSr     OF     FRACTIOIJ^S. 

94      Since.  -::;=,  p  X   -  ,    >t  follows  that,  dmding  by  a 

(juantity  is  equiialeut  to  multiplying  by  its  reciprocal.     Bnt 

c  d 

the  reciprocal   of  a  fraction,    -,,  is   -    (Art.  28);    conse- 

ci  c 

quently,  to  divide  any  quantity  by  a  fraction,  we  invert  the 
terms  of  the  divisor,  and  multij^ly  by  the  resulting  fraction. 
Hence, 

a        c  _  a        d  _  ad 

b    '    d  ~  b        c  ~   bo 

Wlience,  the  folloT\ang  rule  for  dividing  one  fraction  by 
another : 

RULE. 

I.   Reduce  mixed  quantities  to  fractional  forms  : 
II.   Invert  the  terms  of  the  divisor,  a7id  m^idtiply  the 
dividend  by  the  resulting  fraction. 

Note. — ^The  same  remarks  as  were  made  on  factoring 
and  reducing,  under  the  head  of  Multiplication,  are  appli- 
cable in  Division. 

EXAMPLES. 


1.  Divide    a  —  —    by    -  • 
2c      -^     (7 


b         2ae 
a  —  — 


2c  2c 


„  b        f        2ac  —  b       g       2acg  —  bg       . 

Hence,    a r-  -^  =  • — x  '^  =  — ^-z-— ^-   ^'^^ 

2c       ^  2g  f  2cf 


94.  What  iG  the  rule  for  the  division  of  fractions? 


DIVISION      OF      FRACTIONS. 
#2 


103 


2.  DiM(le    -A_2_^Z    by 


2(a  -4-  y)  « 

a  ^    a.2  _   y2 


2(a;  4-  y) 


(a;  -f  y)  (aj  _  y) 


jc  —  y 


^?i5. 


3.  Let    — -    be  divdded  by  —  • 

5  13 

4a.2 

4.  Let    -TT-    be  divided  by  6a;. 


5.  Let    be  divided  by    -— 

6  3 


8. 


Let    be  di\idcd  by   - 

a;  —  1  2 


7.  Let    —    be  divided  by     , 

3  oo 

8.  Let    -~j-    be  divided  by  —j 


^n«. 


91^ 
60 


A  ^» 


Ans. 
Ans. 


g  +  1 
4a; 

2 


a;  —  1 


.        bhx, 

Ans.  — — 

2a 


/l71S. 


X  —  h 


Cc^a; 


Divide  the  folloTiTng  fractions: 


4a;2  -  8a;    ,       a;^  -  4 
0. by    — -— 


10. 


3 

a;*  -  J* 


by 


x^  -\-  hx, 


x^  —  2bx  -{-  b"^    "''      X  -  b 
4a:2 


11.    2a;(a  -h  b)    by 


a  +  6 


y  a;  -  1 

a^  —  ax   .       3(c  —  a) 


13. 


be  -^  bx    ^^^    4(a  -h  a;) 


J  4^       v^ 

a;  H-  2     "^ 


Ans.  X  -{-  --■    / 

(a  +  hy 


Ans 


2x 


Ans.  ^ ^r—^-.   ^ 

yd 

.  Zb(c^  —  a;2) 


104  BLEMENTAIir      ALGEBEA. 

a  —  X    .         1   -f  « 


14.    -—    by       ^ 
1  —  a?      ^    a  +  X 

2x1/ 


15     x^    by    X 


X  +  y 


Ans, - 

1  —  x^ 


Ans.  ^^±^. 


_     b  —  3a  Qa  —  2b 

16.        ^   .        Dy 


2ab 


b^  -  2ab 


x^y  y       X 


Ans, 


A71S, 


2a  —  b 

4a 


x^—  y^ 


X8./C^+l+^^V^  +  ifl 


Ans.   m  -\ 1. 

m 


IP  (x  +  f^)  by  (i  -4^)'    ^^'  y- 

\         1  +  xyj      ''     \  1  -h  ccy/  ^ 


^       20.  /i±-?^ -f  ?)    by    (^±i^^-4-) 


•  An&    1 


BQUATIOMB     OF     THE     FIRST     DEGREE.        106 


CHAPTER   V. 

EQUATIONS     OP     THE     FIRST     DEGREE. 

95.  An  Equation  is  tlie  expression  of  equality  between 
two  quantities.    Thus, 

X      =      b     +     Cy 

is  an  equation,  expressing  the  fact  that  the  quantity  x,  is 
equal  to  the  sum  of  the  quantities  b  and  c, 

96.  Every  equation  is  composed  of  two  parts,  connected 
by  the  sign  of  equality.  Tliese  parts  are  called  members : 
the  part  on  the  left  of  the  sign  of  equality,  is  called  the^rs^ 
member ;  that  on  the  right,  the  second  member.  Thus,  in 
the  equation, 

X  +  a  =  b  —  Cy 

a;  -h  a  is  the  first  member,  and  b  ^  c,  the  second  member. 

97.  An  equation  of  the  Jirst  degree  is  one  which  involves 
only  the  first  power  of  the  unknown  quantity ;  thus, 

6a;  +  3a;  —  5   =   13;   (1  ) 
and  ax  -\-  bx  -\-  c  =    d;    (2) 

are  equations  of  the  first  degree. 

95.  What  IP  an  equation? 

96.  Of  how  many  parts  is  every  equation  composed?    How  are  the 
partfl  connected  ?    What  are  the  parts  called  ?    What  is  the  part  on  the   ■ 
left  called?    The  part  on  the  right? 

97.  What  ip  an  equation  of  the  first  degree 

5* 


ICG  ELEMENTARY      ALGEBRA. 

98.  A  NUATERiCAL  EQUATION  is  ODG  in  Trhich  the  coefR. 
cients  of  the  unknown  quantity  are  denoted  by  numbers. 

99.  A  LITERAL  EQUATION  is  One  in  which  the  coefficients 
of  the  unknown  quantity  are  denoted  by  letters. 

Equation  ( 1 )  is  a  numerical  equation ;  Equation  (  2  )  is  a 
literal  equation. 

EQaATIONS     OP    THE     FIRST     DEGREE    CONTAINING     BUT    ONE 
UNKNOWN    QUANTITY. 

1 00.  The  Transfor:mation  of  an  equation,  is  the  opera- 
tion of  changing  its  form  without  destroying  the  equality 
of  its  members. 

101.  An  Axiom  is  a  self-evident  proposition. 

102.  The  transformation  of  equations  depends  upon  the 
following  axioms : 

1.  If  equal  quantities  he  added  to  both  members  of  an 
equation,  the  equality  will  not  be  destroyed. 

2.  If  equal  quantities  be  subtracted  from  both  mewhers 
of  an  equation,  the  equality  will  not  be  destroyed. 

3.  If  both  members  of  an  equation  he  multiplied  by  the 
same  quantity,  the  equality  will  not  be  destroyed. 

4.  If  both  members  of  an  equation  be  divided  by  the  same 
quantity,  the  equality  will  not  be  destroyed. 

5.  like  powers  of  the  two  members  of  an  equation  are 
equal. 

6.  Lilze  roots  of  the  two  members  of  an  equation  are 
equal. 

98.  What  is  a  numerical  equation  ? 

99    What  is  a  literal  equation  ? 
100.  What  is  the  transformation  of  an  equation  ? 
101    What  is  an  axiom? 

102,  Name  the  axioms  on  which  the  transformation  of  an  equatior 
depends. 


OLEABINO     OF      FHACTION8.  107 

103.  Two  principal  transformations  are  employed  in  the 
solution  of  equations  of  the  first  degree:  Clearing  of  frac- 
tions^ and  Transposing, 

CLR-VErNG    OF    FRACTIONS. 

1,  Take  the  equation, 

2a;       3a;       a; 

The  least  common  multiple  of  the  denominators  is  12.  If 
we  multiply  both  members  of  the  equation  by  1 2,  each  term 
will  reduce  to  an  entire  foim,  giving, 

8a;  —  9a;  4-  2a;  =   132. 

Any  equation  may  be  reduced  to  entire  terms  in  the  same 
manner. 

104.  Hence  for  clearing  of  fractions,  we  have  the  fol- 
lowing 

RULE. 

I.   Find  the  least  common  multiple  of  the  denominators  : 
n.   3fultiply  both  members  of  the  equation  by  it,  reduc- 
ing the  fractional  to  entire  terms. 

Note. — 1.  The  reduction  will  be  effected,  if  we  divide  the 
least  common  multiple  by  each  of  the  denominatoi-s,  and 
then  multiply  the  coiTCsponding  numerator,  dropping  the 
denominator. 

2.  The  transformation  may  be  effected  by  multiplying 
each  numerator  into  the  product  of  all  the  denominators 
except  its  ovnti,  omitting  denominators. 

108.  How  many  transformations  Arc  employed  in  the  solution  of  equa- 
tions of  the  first  degree?     What  are  they? 

104.  Give  the  rule  for  clearing  an  equation  of  fractions  ?  In  what  three 
ways  may  the  redtiction  be  cfTocted? 


108  ELEMENTARiT      ALGEBEA. 

3.  The  transformation  may  also  be  effected,  by  multiplying 
both  members  of  the  equation  by  any  multiple  of  the  de- 
nominators, 

EXAMPLES. 

Clear  the  following  equations  of  fractions: 

1.  ^  -h  ^  —  4   =   3.         Ans.  1x  -{-  5x  —  140  =    105 
5         7 

2.  t;  -\-  -  —  ~  =   S.        Ans.  9x  +  6x  —  2x  =  432. 
b        9        27 


3. 

X          X          X           X 

2  +  3-9+Ii   =   ^°- 

Ans.   18a;  -f  12a;  —  4a;  +  3a5  r= 

720. 

4. 

1*        o*        d* 

-  +  z  —  -   =   4.     Ans.  14a;  -j-  10a;  —  3oa;  = 

280. 

5. 

-  —  -  -h  -   =    15.     Ans.  15a;  —  12a;  -}-  10a;  = 
4        5b 

a;  —  4        a;  —  2         5 

900 

6. 

3                 6        ~   3 

Ans.   —  2a;+8  —  a;-|-2   - 

=   10 

7. 

X                       3 

\     A    —        ,       An"     'it*    I     fin         "Ot»  —   0 

-  3a;. 

3  —  aj                 5 

8. 

?_?  +  ?+?=   12, 
4        6  ^  8  ^  9 

^715.  18a;  —  12a;  -f  9a;  +  8a;  = 

864. 

9. 

-0. 

^  -  1  4-  /  =  5^.         Ans.  ad  -  bo  +  bdf  = 

ax       2c'^x   ,    ,           4^c2a;       5a^   .    26^ 

-T- 7-  +  4a  =   — z -Tz-  -\ 3&. 

b         ab                      a^          b^         a 

bdg 

The  least  common  multiple  of  the  denominators  is  a^b^ 
a*bx—  2a^bc^x  +  4a*b^  =   4b^c^x  —  5a«  -j-  la^b'^c'^  -  ^a^bK 


TBAN8P06INO.  109 


rBANSPOsmo. 

105.  TfiANSPOsrnoN  is  the  operation  of  changing  a  terra 
from  one  member  to  the  other,  without  destroying  the 
equality  of  the  members. 

) .  Take,  for  example,  the  equation, 

6aj  —  6  =  8  H-  2aj. 

I^  in  the  first  place,  we  subtract  2a  from  both  members- 
the  equality  will  not  be  destroyed,  and  we  have, 

6a;  —  6  —  2a;  =  8. 

Whence  we  see,  that  the  term  2a;,  which  was  additive  in 
the  second  member,  becomes  subtractive  by  passing  into 
the  first. 

In  the  second  place,  if  we  add  6  to  both  members  of 
the  last  equation,  the  equality  will  still  exist,  and  we  have, 

5a;  —  6  —  2x  +  6   =  8  +  6, 

or,  since  —  6  and  -f  6  cancel  each  other,  we  have, 

5a;  —  2a;  =  8  4-  6. 

Hence,  the  term  which  was  subtractive  in  the  first  member, 
passes  into  the  second  member  with  the  sign  of  addition. 

106.  Therefore,  for  the  transposition  of  the  terms,  we 
have  the  follo^ving 

RULE. 

Any  term  may  be  transposed  from  one  member  of  an 
equation  to  the  other,  if  the  sign  be  changed, 

106.  What  is  transposition  ? 

106.  What  is  the  rule  for  the  tranHpoeitioD  of  the  terms  of  an  equation? 


110         ELEMENTARY   ALGEBRA. 
EXAMPLES. 

Transpose  the  unknown  terms  to  the  first  member,  and 
the  known  terms  to  the  second,  in  the  following : 

1.  3a;  +  6  —  5  =  2a;  —  7.   Ans.  3a;  —  2a;  =  —  Y  -  6  -f  5 

2.  ax  -\-  b  =  d  —  ex,  Ans.   ax  -\-  ex  =  d  ~  h 

3.  4a;  —  3   =   2a;  +  5.  Ajis.    4a;  —  2a;  ~   5  +  3, 

4.  9a;  +  c  —  ca;  —  d.         A?is.    9a;  —  cjb  =   —  d  —  e, 

5.  ax  -i-  f  =z  dx  -{-  b.  Ans.  ax  —  dx  —  b  —  f. 

6.  Qx  —  c  =   —  ax  -\-  b.         Ans.  Qx  -\-  ax  =  b  -{-  c. 


SOLUTIOl^      OF      EQUATIONS. 

lOT.  The  Solution  of  an  equation  is  the  operation  of 
finding  such  a  value  for  the  unknown  quantity,  as  mil 
satisfy  the  equation ;  that  is,  such  a  value  as,  being  sub- 
stituted for  the  unknown  quantity,  mil  render  the  two  mem- 
bers equal.     This  is  called  a  root  of  the  equation. 

A  Hoot  of  an  equation  is  said  to  be  verified^  when  being 
substituted  for  the  unknown  quantity  in  the  given  equation, 
the  two  members  are  found  equal  to  each  other. 

1.  Take  the  equation, 

|_4   =  1^^4-3. 

Clearing  of  fractions  (Art.  104),  and  performing  the  operar 
lions  indicated,  we  have, 

12a;  —  32   =  4a;  —  8  +  24. 

107.  When  is  the  solution  of  an  equation?  What  is  the  found  value 
of  the  unknown  quantity  called  ?  When  is  a  root  of  an  equation  said  to 
be  verified. 


SOLUTION      OF      EQUATIONS.  Ill 

Transposing  all  the  unknown  terms  to  the  first  member, 
and  the  known  terms  to  the  second  (Art.  IOC),  we  have, 

1205  -  4a;  =   -  8  +  24  -h  32. 

Reducing  the  terms  in  the  two  members, 

8iB  =  48. 

Dividing  both  members  by  the  coefficient  of  Xy 

48 
X  =  -  =  6. 

VERITICATIOX. 

3X6        ,  4(6  -  2)    ,    ^ 

-2--^=  8—  +  ''    or, 

+  9  —  4   =  2  +  3   =   5. 

Hence,  6  satisfies  the  equation,  and  therefore,  is  a  root. 

108.  By  processes  similar  to  the  above,  all  equations  of 
the  first  degree,  containing  but  one  uukno>vn  quantity,  may 
be  solved. 

BULB. 

I.  Clear  the  equation  effractions^  and  perform  all  tho 
indicated  operations  : 

II.  Transpose  all  the  unknown  terms  to  the  first  member, 
and  all  the  known  terms  to  the  second  member : 

in.  Reduce  all  the  terms  in  the  first  member  to  a  single 
term,  one  factor  of  which  will  be  the  unknown  quantity, 
(fiyrl  fhn  other  factor  will  be  tJie  algebraic  sum  of  its  coeffu- 
(■(■  /I  '■• : 

IV.  Divide  both  members  by  the  coefficient  of  the  unknown 
quantity  :  the  second  member  will  tJien  be  the  value  of  t/ie 
wiknown  quantity. 

108.  Give  the  rule  for  aoIviDg  equations  of  the  fir?t  degree  with  ouf 
tinkDown  quantity. 


112  ELEMENTARY      ALGEBRA. 


EXAMPLES. 

1.  Solve  the  equation, 

5x 
12 

4x 

y 

13 

_   1 
""   8 

13aj 

• 

6 

Clearing  of  fractions, 

10a; 

—  32a; 

— 

312   = 

21  - 

52a;. 

By  transposing, 

10a; 

—  32a; 

+ 

52a;  = 

21  4-  312. 

By  reducing. 

30a;   = 

383; 

u  ^^^  111 

hence,  a;  =   —   =  —  =  11.1; 

a  result  which  may  be  verified  by  substituting  it  for  x  in 
the  given  equation. 

2.  Solve  the  equation, 

(3a  —  x)  {a  —  b)  +  2ax  —  4l{x  +  a). 

Performing  the  indicated  operations,  we  have, 

3a2  _  ax  —  Sab  +  6a;  +  2<7a;  =  46a;  +  4a5. 

By  transposing, 

—  ax  -{-  bx  -\-  2ax  —  46a;  —  4ab  +  Sab  —  Sa^, 

By  reducing,  ax  —  36a;  m  lab  —  So?- ; 

Factoring,  (a  —  36)a;  —  lab  —  Sa^. 

Dividing  both  members  by  the  coefficient  of  x, 

lab  -  3^2 

X  = — — . 

a  —  36 

3.  Given    3a;  —  2  +  24  =  31    to  find  x,     Ans.   a;  =  3. 

4.  Given    a;  4-  18   =   3a;  —  5   to  find  x     Ans.  a-  =r  11^ 


80LDTI0N      OF      EQUATIONS.  U3 

6.  Given    6  —  2a;  -f  10   =   20  —  3aj  —  2,  to  fiud  x. 

Arts,  a;  =  2. 

6.  Given    a;  +  ^a;  4-  i«  =   H,  to  find  a?,     xins.  a:  =  6, 

7.  Given    2aj  —  ^a;  +  1    =  5a;  —  2,   to  find  x, 

Ans,   X  =   ij 

Solve  the  fbllowing  equations: 

a  ,  ,  6  —  Sa 

8.  3ax  +  2  -  3  =  &«  -  «.  Ans.   x  =  ^^^TYb' 

9.  ^^  +  f  =  20  -  ^-=-^-  ^1«5.   X  =  23}. 

2  3  2  * 

,^2;4-3,a;         ,        a:  —  5  , 

10.  -^  +  3  =  ^  — r~'  ^''^*  "^^  =  ^^' 

,,     25       3a;    .  4x       „  . 

11.    -  _  —■  +  a;  =   —  -  3.  -4;/5.   a;  =  4. 

4  2  o 

,^     3aaj       2bx        .         ^  .  ^  0^7/*+  4c</ 

13.   ^^  _^4^_Cj4)=  ,oa  +  11*. 

3  o  Z 

Am.  X  =  25a  +  245. 


_fi£5'_(i±£j+ 11  =  0. 


14.  ^  -V"  .  V.4"  ;  rz-f.  ^  =  o.    ^««.  a  =  12. 

1 Z 

,^     a  4-  c  .   a  -  c  2^>2  a^-  5* 

16.    — ; =  -z r         Ans.   X  =  

a  +  X       a  —  X        a^  —  x^  c 

^     Qax  —  h       Sh  —  c         ,        , 

10.      z r =    4—0. 

1  2 

56  -h  96  -  7c 

^1715      X    ~z 

16a 

-      aj       05  —  2       a;         13  , 

17     ---^-  +  -   =-.  a,«    «  =  10. 


114  ELEMENTARY      ALGEBRA. 


18.    5_f  !-?_?-/. 
a       he       d       -^ 

.  ahcdf 

Ans.   X 


bed  —  acd  +  abd  —  ahc 


Note. — ^Wliat  is  the  numerical  value  of  aj,  when   a  =  1 , 
5=2,    c  =  3,    ^  =  4,    and  /  =  6  ? 


19.    ?  _  I  _  ^3  =   _  1211.  Ans.   X  =  l^, 

Sx  —  5        4x  —  2  ,    ,  . 

20     JK  ^ ^_ 1 —  =  X  -\-  1,       Ans.  a  =  6, 

lo  11 

Sy  cC  Q! 

21.  a  +  -  +  -  —  -  =   203  —  43.  A7is.   x  =  60. 

4        5        6 

o^     «         4cB  —  2         3a;  —  1  , 

22.  2a; —  =  — Ans.   x  =  S. 

o  2 

««     «      .    ^a;  —  c?  .  .  Sa  -\-  d 

23.  3a;  4-,  — - —  =  x  -\-  a.  Ans.  x  =         ,        • 

ax  —  b       a  __  hx       hx  —  a 
'  4        "^  3   ~  T  ^3 

3d 

-4?25.    X    = 


3a—  26 


4a;  20  -  4a;         15  .  „  2 

25. = A71S.   X  =  3— -• 

6  —  a;  a;  a;  11 

2a;  +  1  _  402  —  3a;  _  ^  _  471  —  Qx 
'         29  12  ~  2 

A?is.   a;  =  72, 

^^     (a  -{-h)(x  —  b)       „  4«J  -  52    -  a^_  i,,^ 

21      -  ^-^ ^  -  3a  =  —7 2a;  -] , 

a  —  h  a-\-b  b 

a*  +  Sa^b  -f  4a2J2  _  q^J)^  ^  26" 


-4'i.<f.   a; 


2^(2a2  ^  ab  -  b"^) 


PROBLEMS.  116 


PROBLEMS. 

109.  A  Pr.OELKir  is  a  question  jroposecl,  requiring  a 
solution. 

Tlie  Solution  of  a  problem  is  the  operation  of  finding  a 
quantity,  or  quantities,  that  will  satisfy  the  given  conditions. 

The  solution  of  a  problem  consists  of  two  parts : 

I.  The  STATEMENT,  whicJi  consists  in  expressing^  algehror 
icaUy^  the  relation  between  the  known  and  the  required 
qita7itities. 

n.  The  SOLUTION,  ichich  C07isists  in  finding  the-  values 
of  the  unknown  quantitcs^  in  terms  of  those  which  are 
known, 

Tlie  statement  is  made  by  representing  the  unknown 
quantities  of  the  problem  by  some  of  tlie  final  letters  of  the 
alphabet,  and  then  operating  upon  these  so  as  to  comply 
"with  the  conditions  cf  the  problem.  The  method  of  stating 
problems  is  best  learned  by  practical  examples. 

1.  What  number  is  that  to  which  if  5  be  added,  the  sum 
will  be  equal  to  9  ? 

Denote  tlie  number  by  x.     Then,  by  the  conditions, 

JB  +  5   =  9. 
This  is  the  statement  of  the  problem. 

To  find  the  value  of  jc,  transi)ose  5  to  the  second  member; 
then, 

X   =  9  —  5  =  '4. 

Tliis  is  the  solution  of  the  equation. 

VERIFICATION. 

aj  +  5  =  0. 

109.  Wliat  is  a  problem?  "What  is  the  Folution  of  a  problem?  Of 
how  many  parte  does  it  consist  *  What  are  they  ?  What  is  the  state- 
me&t  ?    What  is  the  solution  ? 


116  ELEMENTART      ALGEBRA. 

2,  Find  a  number  such  that  the  sum  of  one-half,  one-tliird, 
and  one-fourth  of  it,  augmented  by  45,  shall  be  equal  to  448 

Let  the  required  number  be  denoted  by       x, 

Tlien,  one-half  of  it  will  be  denoted  by       - , 

one-third        "  "  by 


one-fourth       "  "  by 

and,  by  the  conditions, 


X 

3' 

X 

4' 


1  +  1  +  1  +  45   =  448. 
This  is  the  statement  of  the  problem. 

Clearing  of  fractions, 

6a;  +  405  -I-  3a;  +  540  =  5376  , 
Transposing  and  collecting  the  unkno-wTi  terms, 
13a;  =  4836; 

4836 
hence,  x  =  -— -   =  372. 

lo 

VERIFICATION. 

?^  +  ?!?  +  ?1?  +  45   =   186  +  124  -^  93  ^    45   =   448> 
2  o  4 

3.  What  number  is  that  whose  third  part  exceeds  its 
fourth  by  16? 

Let  the  required  number  be  denoted  by  x.     Then, 

-X  —    the  third  part, 
3 

-X  z=   the  fourth  part 


PB0BLBM8.  117 

and,  hj  the  conditions  of  the  problem, 

~x  '-  -X  =  16. 
3  4 

This  i£  the  statement.    Clearing  of  fractions, 

4a;  —  3aj  =  192, 

and  hence,  x  =  192. 

VERIFICATION. 

192        192 

i^  -  i^   =  64  -  48  =  16. 
3  4 

4.  Divide  llOOO  between  A^  B^  and  (7,  so  that  A  shall 
have  $72  more  than  J?,  and  C  llOO  more  than  A. 

Let  X  denote  the  number  of  dollars  which  B  received. 

Then,  x  =    B''s  number, 

JB  4-  72  =  A'*s  number, 
and,  jc  +  172    =    (7's  number; 

and  their  sum,  Zx  +  244    =    1000,  the  number  of  dollars. 

This  is  the  statement.    By  transposing, 
3aj  =   1000  -  244   =   756  ; 

and,  X  =    — -   =  252  =  B's  share. 

3 

Hence,  a  +  72  =  252  +  72  =  324  =  A's  share, 
and,  a;  -f  172    =  252   +   172  =  424  =    C'«  share. 

VERinCATION. 

252  +  324  4-  424  =  1000. 

6.  Out  of  a  cask  of  wine  which  had  leaked  away  a  third 
part,  21  gallons  were  afterwards  dra^m,  and  the  cask  being 
then  gauged,  appeared  to  be  half  full :  how  much  did  it 
hold? 


118  ELEMENTARY      ALGEBRA. 

Let      X  denote  the  number  of  gallons. 

gj 

Tlien,  -   =    the  number  that  had  leaked  away. 

85 

and,         -  +  21    =    what  had  leaked  and  been  drawn. 

d 

OR  CC 

Hence,  by  the  conditions,   -  +  21    =    -  • 

o  ^ 

This  is  the  statement.     Clearing  of  fractions, 

2x  +  126   =  3x, 
and,  —  X  =z   —  126  ; 

and  by  changing  the  signs  of  both  members,  which  does  not 
destroy  their  equality  (since  it  is  equivalent  to  multi^Dlying 
both  members  by  —  1),  we  have, 

X  —  126. 

VERIFICATION. 

1|5  +  21    =   42  +  21    =  63   =  ij-^ 

6.  A  fish  was  caught  whose  tail  weighed  g  lbs.,  his  head 
weighed  as  much  as  his  tail  and  half  his  body,  and  his  body 
weighed  as  much  as  bis  head  and  tail  together :  what  was 
the  weight  of  the  fish  ? 

Let  2x  =  the  weight  of  the  body,  ui  pounds. 

Then,      9  +  cc  =  weight  of  the  head ; 

and  since  the  body  weighed  as  much  as  both  head  and  tail| 

2aj  =  9  -}-  9  +  JC, 
which  is  the  statement.    Then, 

2x  —  X  =  18,   and  x  =  18. 


PROBLEMS.  119 

Hence,  we  have, 

2x  =  SQlb.  =  weight  of  the  body, 

9  +  X  =  21lb.  —  weight  of  the  head, 

9/^.  =  weiglit  of  the  tail ; 

hence,  I2lb,  =  weight  of  the  Mu  • 

7.  The  Slim  of  two  numbers  is  67,  and  their  difference  19 . 
what  are  the  two  numbers  ? 

Let      X  denote  the  less  number. 

Tlien,  X  +  19  z=  the  greater;  and,  by  the  conditions, 

2aj  +  19  =  67. 

This  is  the  statement.     Transposing, 

2a;  =  67  —  19  =  48 ; 

AH 

hence,  a;  =  —  =  24,   and  a;  +  19  =  43. 

VERTFICATION, 

43  +  24  =  67,   and   43  —  24  =  19. 

ANOTHEK    SOLUnOX. 

Let  X  denote  the  greater  number. 

Then,        x  —  1 9   will  represent  the  less, 
and,  2a:  —  19    =    67;    whence    2a;  =  67  -f  19. 

r^  n  86 

Therefore,  x  =  —  =43; 

and,  consequently,    a;  —  19  =  43  —  19  =  24. 

GENERAL    SOLUTION    OF    THIS    PROBLEM. 

The  eum  of  two  numbers  is  8,  their  difference  is  di  what 
are  the  two  numbers  ? 


120  ELEMENTARY      ALGEBRA. 

Let  X  denote  the  less  number. 

Tlien,         X  -{-  d  will  denote  the  greater, 
and  2x  +  d   =  s,   their  sum.     WTience, 

_  s  —  d  _   s       d  ^ 

^  ~       2        ~  2  ~  2* 
and,  consequently, 

2        2  2        2 

As  these  two  results  are  not  dependent  on  particular 
values  attributed  to  s  or  <?,  it  follows  that : 

1.  The  greater  of  two  numbers  is  equal  to  half  their  sum, 
plus  half  their  difference  : 

2.  The  less  is  equal  to  half  their  sum,  m,inus  half  their 
difference. 

Thus,  if  the  sum  of  two  numbers  is  32,  and  their  differ- 
ence 16, 

32       16 
the  greater  is,        -—  +  -—=  16  +  8  =  24  ;    and 

^  ji 

,     ,                         32        16 
the  less,  — —=16  —  8=8. 

VERIFICATION. 

24  4-  8   =  32 ;   and   24  —  8   =   16. 

8.  A  person  engaged  a  workman  for  48  days.  For  each 
day  that  he  labored  he  received  24  cents,  and  for  each  day 
that  he  was  idle,  he  paid  12  cents  for  his  board.  At  the 
end  of  the  48  days,  the  account  was  settled,  when  the  laborer 
received  504  cents.  Required,  the  number  of  working  days, 
and  the  number  of  days  he  was  idle. 

If  the  number  of  working  days,  and  the  number  of  idle 
days,  were  known,  and  the  first  multiplied  by  24,  and  the 


PROBLEMS.  121 

second  by  12,  the  difference  of  these  products  would  be 
604.  Let  us  indicate  these  operations  by  means  of  algebraic 
signs. 

Let  X  denote  the  number  of  working  days. 

Then,      48  —  SB    =    the  number  of  idle  days, 
24  X  aj   =   the  amount  earned, 
ai;d,       12(48  —  x)  =   the  amount  paid  for  board. 

Then,  24a;  -  12(48  —  a)   =  604, 

what  was  received,  which  is  the  statement. 

Then,  perfonning  the  operations  indicated, 

24aj  —  6V6  +  12a;  =     604, 
or,  36a;  =  504  4-  576   =  1080, 

and,  X  =  -— -  =  30,  the  number  of  working  days; 

36 

whence,       48  —    30     =18,  the  number  of  idle  days. 

VKKIFICATION. 

Thirty  days'  labor,  at  24  cents  )^  ^  ^4  =  720  cents. 
a  day,  amounts  to ) 

And  18  days*  board,  at  12  cents  )  ,^       ,„         ^,^ 
,                 \   ^          '                      M8  X  12  =  216  cents. 
a  day,  amounts  to )  

The  difference  is  the  amount  received    ....   604  cents. 


GKNEKAL    SOLUTION, 

This  problem  may  be  made  general,  by  denoting  the  whole 
number  of  working  and  idle  days,  by  n  ; 

The  amount  received  for  each  day's  work,  by  a ; 

The  amount  paid  for  boai*d,  for  each  idle  day,  by  b ; 

And  what  was  due  the  laborer,  or  the  balance  of  th« 
aoooont,  by  c. 
0 


122  ELEMENTARY       ALGEBRA. 

As  before,  let  the  number  of  working  days  be  denoted 
by  cc. 

The  number  of  idle  days  will  then  be  denoted  by  7i  —  x. 

Ilcnce,  what  is  earned  will  be  (expressed  by  ax^  and  the 
sum  to  be  deducted,  on  account  of  board,  by  b{n  —  x). 

The  statement  of  tlie  proljlem,  therefore,  is, 

ax  —  h[7i.  —  x)  =  c. 
Pel  forming  indicated  operations, 

ax  —  bn  -\-  bx  =  c,    or,    (a  +  b)x  =  c  +  b?i 

whence,  .t  = r    —    number  of  workin 2^  days; 

_                                      c-^b?i       an-\-b7%—c—bn 
and,  n  ^  X  =  n —7-  = —- , 

or,  n  —  X  =  7-  =    number  of  idle  days. 

'  a  -\-  b  -^ 

Let  us  suppose  7^  =  48,  a  —  24,  5  =  12,  and  c  =  504  ; 
these  numbers  will  give  for  x  the  same  value  as  before 
found. 

9.  A  person  d}ang  leaves  half  of  his  property  to  his  wife, 
one-sixth  to  each  of  tAvo  daughters,  one-twelfth  to  a  servant, 
and  the  remaining  $600  to  the  poor ;  what  was  the  amount 
of  the  property  ? 

Let  X  denote  the  amount,  in  dollars, 

Then,       -    =      what  he  left  to  his  wife, 

X 

-    =      w^hat  he  left  to  one  daughter, 

Q.J*  gj 

and,  —  =  -  what  he  left  to  both  daughters, 

6  3 

X 

alfio  —  =       what  he  left  to  his  servant, 

12 

and,         $600   —       'ciiat  he  left  to  the  poor. 


PHOBLEMB.  123 

Then,  by  the  conditions, 

-  -h  -  +  rr  +  600  =  X,  the  amount  of  the  pi  operty, 

wJiich  g\\  es,  X  =  $7200. 

10.  A  and  B  play  together  at  cards.  A  sits  down  with 
|84,  and  i?  with  $48.  Each  loses  auad  wins  in  turn,  when 
it  appears  that  A  has  five  times  as  much  as  J>,  How  much 
did  A  win  ? 

Jjei  X  denote  the  number  of  dollars  A  won. 
Then,  A   rose  with   84  -h  a;  dollars, 

and  -B  rose  with  48  —  a;  dollars. 

But,  by  the  conditions,  we  have, 

84  +  JB  =  5(48  -  x), 

hence,  84  +  a  =  240  —  5x; 

and,  6a!  =  156, 

consequently,  x  =  26  ;   or  ^  won  $26. 

VERIFICATION. 

84  4-  26  =  110  ;     48  —  26  =  22; 
110  =  5(22)   =  110. 

11.-4  can  do  a  piece  of  work  alone  in  10  days,  J?  in  13 
days ;  in  what  time  can  they  do  it  if  they  work  together  ? 

Denote  the  time  by  x,  and  the  work  to  be  done,  by  1. 
Then,  in 

1  day,  A  can  do  —  of  the  work,  and 

i?  can  do  —  of  the  work ;  and  in 
13 

X 

X  days,  A  can  do  —    of  the  work,  and 

^  can  do   —   of  the  work. 
13 


124  ELEMENTARY      ALGEBRA. 

Hence,  by  tlie  conditions, 

— -  +  —  =  1    "v\-hich  gives,    13aj  +  10a;  =::   130; 
10         13 

130 
hence,        23a;  =  130,     x  =  — —  =  5|f  days. 

12  A.  fox,  pursued  by  a  hound,  has  a  start  of  60  of  his 
own  leaps.  Three  leaps  of  the  hound  are  equivalent  to  7  of 
the  fox ;  but  while  the  hound  makes  6  leaps,  the  fox  makes 
9 :  how  many  leaps  must  the  hound  make  to  overtake  the 
fox? 

There  is  some  difficulty  in  this  problem,  arising  from  the 
different  units  which  enter  into  it. 

Since  3  leaps  of  the  hoimd  are  eqjial  to  7  leaps  of  the  fox, 

7 
1  leap  of  the  hound  is  equal  to  -  fox  leaps. 
*  3 

Since,  while  the  hound  makes  6  leaps,  the  fox  makes  9, 

9  3 

while  the  hound  makes  1  leap,  the  fox  will  make  - ,  or  - 

leaps. 

Let  X  denote  the  numhcr  of  leaps  which  the  hound  makes 
before  he  overtakes  the  fox ;  and  let  1  fox  leap  denote  the 
unit  of  distance. 

Since  1  leap  of  the  hound  is  equal  to  -   of  a  fox  leap,  x 

1  .      . 

leaps  tvtU  be  equal  to  -a:  fox  leaps ;  and  this  will  denote  the 
3 

distance  passed  over  by  the  hound,  in  fox  leaps. 

Q 

Since,  wHle  the  hound  makes  1  leap,  the  fox  makes  - 

3 
leaps,  wliile  the  hound  makes  x  leaps,  the  fox  makes  -a;  leaps ; 

Ji 

and    this    added    to    60,    his    distance    ahead,  will    give 

g 

~x  -{-  60,   for  the  whole  distance  passed  over  by  the  fox 

2i 


PROBLEM  8.  125 


Hence,  from  the  conditions, 


7  3 

-a;  =  -a;  +  GO ;  wnenco, 

3  2 

14a;  =  9a;  +  360; 

X  =  72. 


The  hound,  therefore,  makes  72  leaps  before  overtakmg 
le  fo 
leaps. 


tlie  fox;  in  the  same  time,  the  fox  makes  72  x  -  =  108 


VERIFICATION. 

108  4-  60  =  168,    whole  number  of  fox  leaps, 
72  X     ^  =  168. 

o 

13.  A  father  leaves  his  property,  amounting  to  $2520,  to 
four  sons,  A^  J5,  (7,  and  D.  (7  is  to  have  $360,  i?  as  much 
as  C  and  D  togetlier,  and  A  twice  as  much  as  i?,  less  $1000 : 
how  much  do  -4,  i?,  and  D  receive  ? 

Aiu,   ^,$760;  ^,$880;  2>,  $520. 

14.  An  estate  of  $7500  is  to  be  divided  among  awddow, 
two  sons,  and  three  daughters,  so  that  each  son  shall  receive 
twice  as  much  as  each  daughter,  and  the  widow  herself  $500 
more  than  all  tlie  children :  what  was  her  share,  and  what 
the  share  of  each  child  ? 

(  Widow's  share,     $4000. 

Am,    \  Each  son's,  1000. 

(  Each  daughter's,      500. 

15.  A  company  of  180  persons  consists  of  men,  women, 
and  cliildren.  The  men  are  8  more  in  number  than  the 
women,  and  the  children  20  more  than  the  men  and  women 
together :  how  many  of  each  sort  in  the  company  ? 

Al^s.   44  men,  36  women,  100  children. 


126  ELEMENTARY      ALGEBRA. 

16.  A  father  divides  $2000  among  five  sous,  so  that  each 
elder  should  receive  $40  more  than  his  next  younger  bro« 
ther  :  what  is  the  share  of  the  youngest?  Afis.   $320. 

17.  A  purse  of  $2850  is  to  be  divided  among  three  per 
sons,  A,  B^  and  C.  A's  share  is  to  be  to  B''s  as  C  to  11, 
and  C  is  to  have  $300  more  than  A  and  B  together :  what 
is  each  one's  share?  A's,  $450  ;  B's,  $825  ;  C's,  $1575. 

18.  Two  pedestrians  start  from  the  same  point  and  travel 
in  the  same  direction ;  tlie  first  steps  twice  as  far  as  the 
second,  but  the  second  makes  5  steps  while  the  first  makes 
but  one.  At  the  end  of  a  certain  time  they  are  300  feet 
apart.  Now,  allowing  each  of  the  longer  j)aces  to  be  3  feet, 
bow  far  will  each  have  traveled  ? 

Ans.    1st,  200  feet ;  2d,  500. 

19.  Two  carpenters,  24  journeymen,  and  8  apprentices 
received  at  the  end  of  a  certain  time  $144.  The  carpenters 
received  $1  per  day,  each  journeyman,  half  a  dollar,  and 
each  apprentice,  25  cents :  how  many  days  were  they  em- 
ployed ?  Alls.    9  days. 

20.  A  capitalist  receives  a  yearly  income  of  $2940  ;  four- 
fifths  of  his  money  bears  an  interest  of  4  per  cent.,  and  the 
remamder  of  5  per  cent. :  how  much  has  he  at  interest  ? 

Ans.    $70000. 

21.  A  cistern  containing  60  gallons  of  water  has  three 
unequal  cocks  for  discharging  it ;  the  largest  will  empty  it 
in  one  hour,  the  second  in  two  hours,  and  the  third,  in  three: 
in  what  time  will  the  cistern  be  emptied  if  they  all  run  to- 
gether ?  Ans.    32^\  min. 

22.  In  a  certain  orchard,  one-half  are  api)le  trees,  one- 
fourth  })each  trees,  one-sixth  plum  trees;  there  are  also,  120 
clicriy  trees,  and  80  pear  trees:  how  many  trues  in  tha 
orchard?  Ans.  2400. 

23.  A  farmer  being   asked   how  many  sheep  he   had^ 


rfiOBLEMS.  127 

answered,  that  he  had  them  in  five  fields ;  in  the  Ist  he  had 
J,  in  llic  2d,  j,  in  the  3d,  |,  and  in  the  4th,  j^,  and  in  llie 
5th,  450  :  how  many  had  he  ?  A}is.  1200. 

24.  My  horse  and  saddle  together  are  worth  |132,  and 
the  horse  is  worth  ten  times  as  much  as  the  saddle:  wlmi 
is  tlie  value  of  the  horse  ?  Ans.   |120. 

25.  Tlie  rent  of  an  estate  is  this  year  8  per  cent,  greater 
than  it  was  last.  This  year  it  is  $1890;  what  was  it  last 
year?  Ans.   $1750. 

26.  Wliat  number  is  that,  from  which  if  5  be  subtracted, 
§  of  the  remainder  will  be  40  ?  Ans.   05. 

27.  A  post  IS  \  in  the  mud,  i  in  the  water,  and  10  feet 
above  the  water :  what  is  the  whole  length  of  the  post  ? 

Ans,   24  feet. 

28.  After  paying  ^  and  ^  of  my  money,  I  had  66  guineas 
left  in  my  purse  :  how  many  guineas  were  in  it  at  first  ? 

Ans.   120. 

29.  A  person  was  desirous 'of  giving  3  pence  apiece  to 
some  beggars,  but  found  he  had  not  money  enough  in  his 
pocket  by  8  pence;  he  therefore  gave  them  each  2  pence 
and  had  3  pence  remauiing :  required  the  number  of  beg- 
gars. Ans.    11. 

30.  A  person,  in  play,  lost  |  of  his  money,  and  then  won 
3  sliillings ;  after  which  he  lost  ^  of  what  he  then  had  ;  and 
this  done,  found  that  ho  had  but  12  shilUngs  remauiing: 
what  had  he  at  first  ?  Ans.  20s. 

31.  Two  persons,  A  and  i?,  lay  out  equal  sums  of  money 
in  trade;  A  gains  $120,  and  J3  loses  $87,  and  A''s  money  is 
tlien  double  of  JPs :  what  did  each  lay  out?        Ajis.   $300. j 

82.  A  person  goes  to  a  tavern  with  a  certain  sum  of 
money  m  his  pocket,  where  he  spends  2  shillings:  he  then 
borrows  as  much  money  as  he  had  left,  and  going  to  another 
tavern,  he  there  spends  2  shillings  also;   then  borrowmg 


128  ELEMENTARY      ALGEBRA. 

again  as  much  money  as  was  left,  he  went  to  a  third  tavern, 
where  Ukewise  he  spent  2  shillings,  and  borrowed  as  much 
as  he  had  left :  and  again  spending  2  shillings  at  a  fourth 
lavem,  he  then  had  nothing  remaining.  What  had  he  at 
lirst  ?  Ans,   35.  9c?, 

33.  A  tailor  cut  19  yards  from  each  of  three  equal  pieces 
of  cloth,  and  17  yards  from  another  of  the  same  length, 
and  found  that  the  four  remnants  were  together  equal  to 
142  yards.     How  many  yards  in  each  piece?  Ans,   54. 

34.  A  fortress  is  garrisoned  by  2600  men,  consisting  of 
infantry,  artillery,  and  cavalry.  Now,  there  are  nine  times 
as  many  infantry,  and  three  times  as  many  artillery  soldiers 
as  there  are  cavalry.     How  many  are  there  of  each  corps  ? 

Ans.    200  cavalry;  600  artillery ;  1800  infantry. 

35.  All  the  joumeyings  of  an  individual  amounted  to  2970 
mUes.  Of  these  he  traveled  3^  times  as  many  by  water  as 
on  horseback,  and  2i  times  as  many  on  foot  as  by  water. 
How  many  miles  did  he  travel  in  each  way  ? 

A71S.    240  miles;  840  m.;  1890  ra. 

36.  A  sum  of  money  was  divided  between  two  persons, 
A  and  Ji.  A''s  share  was  to  JB^s  in  the  proportion  of  5  to  3, 
and  exceeded  five-nmths  of  the  entire  sum  by  50.  "What 
was  the  share  of  each?      A7is.   ^'5  share,    450;  ^'5,    270. 

37.  Divide  a  number  a  into  three  such  parts  that  tho 
second  shall  be  n  times  the  first,  and  the  third  m  times  a? 
great  as  the  first. 

a  ,  na         ^      ,  ma 

^^'    I  -^m  +  n*    ^^'   1  +  m  +  n'        '    1  -h  m -h  n 

38.  A  father  directs  that  111 70  shall  be  di\ided  among 
his  three  sons,  in  proportion  to  their  ages.  The  oldest  is 
twice  as  old  as  the  youngest,  and  the  second  is  one-third 
older  :han  the  youngestc     How  much  was  each  to  receive  ? 

Ans.    $270,  youngest;  $360,  second  ;  $540,  oldest 


PROBLEM  8.  129 

39.  Three  regiments  are  to  furnish  594  men,  and  each  to 
fiiruieh  in  proportion  to  its  strength.  Now,  the  strengtli  of 
the  fii-st  is  to  the  second  as  3  to  5  ;  and  that  of  the  second 
to  the  third  as  8  to  7.     How  many  must  each  furnish  ? 

Am.    1st,  144  men ;  2d,  240  ;  3d,  210 

40.  Five  heirs,  -4,  J5,  C,  jO,  and  ^,  arc  to  divide  an  inhtr  - 
itanco  of  $5600.  J5  is  to  receive  twice  as  much  as  Ay  and 
|200  more ;  C  three  times  as  much  as  Ay  less  $400 ;  D  tlie 
half  of  what  £  and  C  receive  together,  and  150  more ;  and 
JS  the  fourth  part  of  what  the  four  others  get,  plus  $475. 
How  much  did  each  receive  ? 

A%  $500;  JB'Sy  1200;   C's,  1100;  D*Sy  1300;  U's,  1500. 

k'  ,%,    1- 

r  is/i  741.  A  person  has  four  casks,  the  second  of  wliich  being 
filled  from  the  nrst,  leaves  the  first  fouj»-seventlis  full.  The 
third  Jjcing  filled  from  the  second,  leaves  it  one-fourth  fulj, 
and  when  the  thiid  is  emptied  hito  the  fourth,  it  is  found  to 
fill  only  nine-sixteenths  of  it.  But  the  first  will  fill  the  third 
and  fourth,  and  leave  15  quarts  remaining.  How  many 
gallons  does  each  hold  ? 

Ans.   1st,  35  gal. ;  2d,  15  gal. ;  3d,  11 J  gal. ;  4th,  20  gal. 

42.  A  courier  having  started  from  a  place,  is  pursued  by 
a  second  after  the  lapse  of  10  days.  The  fii*st  travels  4 
miles  a  day,  the  other  9.  How  many  days  before  the 
second  will  overtake  the  first  ?  Atis,   8. 

43.  A  courier  goes  31^  miles  every  five  hours,  and  is  fol- 
lowed by  another  after  he  had  been  gone  eight  hours.  Tlie 
second  travels  22^  miles  every  three  hours.  How  many 
hours  before  he  will  overtake  the  first  ?  Ans,  42. 

44.  Two  places  are  eighty  miles  apart,  and  a  person  leaves 
one  of  tliem  and  travels  towards  the  other  at  the  rate  of  3j 
miles  per  hour.    Eight  hours  after,  a  person  departs  from 

6* 


130  ELEMENTARY      ALGEBRA. 

the  second  place,  and  travels  at  the  rate  of  5}  miles  per  hour 
How  long  before  they  will  be  together  ? 

Ans.   6  hoTirs, 

EQUATIONS   CONTAINING   TWO   UNKNOWN   QUANTmES« 

1 10.     If  w^e  have  a  single  equation,  as, 

2£C  +  3?/  =  21, 

containing  tw^o  unknown  quantities,  x  and  y,  wo  may  find 
the  value  of  one  of  them  ui  terms  of  the  other,  as. 


21  -3y 

2 


^  =   — — (1.) 


Now,  if  the  value  of  y  is  unknow^n,  that  of  x  will  also  be 
unknown.  Hence,  from  a  single  equation,  contaming  two 
unknown  quantities,  the  value  of  x  cannot  be  determined. 

If  w^e  have  a  second  equation,  as, 

5x  +  41/  =   35, 

we  may,  as  before,  find  the  value  of  a;  in  terms  of  y,  giving, 

35  —  4v  ,     , 

^  =  1-^ (2.) 

Now^,  if  the  values  of  x  and  ?/  are  the  same  in  Equations 
(1 )  and  (  2 ),  the  second  members  may  be  placed  equal  to 
each  other,  giving, 

21  -  32/         35  -  4y  ,^„        ^^  ^^        ^ 
—-^  =   ■ ^  ,    or    105  -  ISy  =   70  -  8y ; 

from  Avhich  we  find,  y  =  5. 

110.  In  one  equation  containing  two  unknown  quantities,  can  yon  find 
the  vaiue  of  e^her  ?  if  you  have  a  second  equation  involving  the  same 
two  unknown  quantities,  can  you  find  their  values  ?  What  are  such  equa- 
tions  called  ? 


ELIMINATION.  131 

Subtituting  tliis  value  for  y  in  Equations  (  1 )  or  (  2  j,  we 
find  X  —  Z.  Sucli  equations  are  called  tSlmultaneoua 
equations.    Hence, 

111.  Simultaneous  Equations  are  those  in  which  the 
values  of  the  unknown  quantity  are  the  same  in  both. 

ELDIINATION. 

113.  ELniiNATioN  is  the  operation  of  combining  two 
equations,  containini^  two  unknown  quantities,  and  deduciug 
tJierefrom  a  single  equation,  containing  but  one. 

Tlicre  are  three  principal  methods  of  eUnuuation  : 

1st.  By  addition  or  subtraction. 

2d.    By  substitution. 

3d.   By  comparison. 
We  shall  consider  these  methods  separately. 

Elimination  by  Additio7i  or  Subtraction, 
1.  Take  the  two  equations, 

3aj  —  2y  =     7, 
8a;  +  2y  =  48. 

If  we  add  these  two  equations,  member  to  member,  we 
obtain, 

11a;  =  55; 

which  gives,  by  dividing  by  11, 

jc  =  5; 

and  substituting  this  value  in  either  of  the  given  equations, 
we  find, 

y  =  *. 


111.  What  are  simultaneous  cquatiou^P 

112.  What  \s  eliiiiinutiou?      Ilow  many  methods  of  elimiDation  ore 
there  f     What  are  they  ? 


132  ELEMENTAEY      ALCJEBEA. 

2.  Again,  take  the  equations, 

8a;  +  2y  =  48, 
3iB  4-  2?/  ==   23. 

If  we  subtract  the  2d  equation  from  the  1st,  we  obtain, 

5a;  =   25; 

which  gives,  by  dividing  by  5, 

a;  =  5; 

and  by  substituting  this  value,  we  find, 

2/  =  4. 

3.  Given  the  sum  ol  two  numbers  equal  to  s,  and  their 
difierence  equal  to  J,  to  find  the  numbers. 

Let  X  =  the  greater,  and  y  the  less  number. 

Then,  by  the  conditions, x  -]-  y   —    s. 

and,       X  —  y   ^  d. 

By  adding  (Art.  102,  Ax.  1), 2a;  =    5  4-  d 

By  subtracting  (Art.  102,  Ax.  2),  .  .  ,  2y  =r:  s  -^  d. 
Each  of  these  equations  contains  but  one  unknown  quantity. 

From  the  first,  we  obtain,       ..,.,.    a;  -^    — ■ — , 

2 

and  from  the  second, ,     V  = - 

These  are  the  same  values  as  were  found  in  Prob.  7,  page 
120. 

4.  A  person  engaged  a  workman  for  48  days.  For  each 
day.  that  he  labored  he  was  to  receive  24  cents,  and  for  each 
day  that  he  was  idle  he  was  to  pay  12  cents  for  his  board. 
At  the  end  of  the  48  days  the  account  was  settled,  when  the 
laborer  received  504  cents.  Required  the  number  of  work- 
ing days,  and  the  number  of  days  he  was  idle. 


ELIMINATION.  133 

Let  X  =     the  numbBr  of  working  days, 

y  =     the  number  of  idle  days. 

Then,  24a;  =     what  he  earned, 

and,  \2y  =.     what  he  paid  for  his  board. 

Tlien,  by  the  conditions  of  the  question,  we  have, 

a;  +      y  =     48, 
and,  24aj  —  12y  =  604. 

This  is  the  statement  of  the  problem. 

It  has  already  been  shown  (Art.  102,  Ax.  3),  that  the  two 
members  of  an  equation  may  be  multiplied  by  the  same  num- 
ber, without  destroying  the  equality.  Let,  then,  the  first 
equation  be  multiplied  by  24,  the  coefficient  of  aj  in  the 
second ;  we  shall  then  have, 

24a;  -f  24y  =  1152 
24a;  —  12y  =     504 

and  by  subtracting,  36y  =     648 

648 
.'.     V  =  =   18. 

^  36 

Substituting  this  value  of  y  in  the  equation, 
24a;  —  12y  =  504,     we  have,     24aj  —  216  =  504; 

vi'hich  gives, 

720 
24aj  -r-.  604  +  216  =  720,     and     x  =^  —^  =  80. 

24 

VERIFICATION. 

jc  +      y  =     48     gives  30  +  18  =     48. 

24a:  —  12y  =   504     gives     24  x  30  -  12  X  18   =  504 


134 


ELEMENTAET      ALGEBKA 


113.     In  a  similar  manner,  either  unkno^^Ti  quantity  may 
be  eliminated  from  either  equation ;  hence,  the  following 


EULE. 


I.  Prepare  the  equations  so  that  the  coefficients  of  the 
quantity  to  be  eliminated  shall  be  numerically  equal: 

II.  If  the  signs  are  unlike^  add  the  equations^  member 
to  member ;  if  alike^  subtract  them^  member  from  mend^er. 


EXAMPLES. 


Find  the  values  of  x  and  y,  by  addition  or  subtraction, 
in  the  following  simultaneo'j.s  equations : 


-  y  = 

+  2x  = 


n 


ibx  ■\-  2y  ■=         37  ) 

^    J  2aj  +  6y  =  42  > 
*   (  8aj  —  6y  =     3  i"    ■ 

3    j  8:«  -  9y  =   1    ) 
(  6a;  —  3y  =  4a;  i 

Q    j  14a;  -  15y  ^   12  ) 

'   (    1x  -{-    8?/  —   37  ) 

1 


Ans.  X  =  2,  y  =  3. 
Ans.  X  =  5^  y  —  Q* 
Ans,  a;  =  4^,  y  —  b\. 
Ans.  a;  =  I,  y  =  i. 
Ans.   jc  =  3,   y  =  2. 


10.  < 


1^  +  3^^ 

1,1 

3^  +  2^   =   61 


Ans. 


6,   y  =  9. 


n.J"  +  §^  =  ' 


A.ns.  <  a;  =  14,   y  =:  IG. 


y 


-  2 


113.  What  is  the  rule  for  elimination  by  addition  or  subtractiou  ? 


ELIMINATION.  135 

12.  Says  A  to  B^  you  give  mo  $40  of  }oiir  money,  ami 
I  sliall  then  liavo  five  times  as  much  as  you  will  have  lell. 
Now  they  both  ha^$120;  how  much  had  each? 

"^  Ans.  Each  had  $60. 

13  A  father  says  to  his  son,  "  twenty  years  ago,  my  ago 
was  four  times  yours;  now  it  b  just  double:"  what  were 
their  ages  ?  A       \  Father's,  CO  years. 

*  (  Son's,       30  years. 

14.  A  father  divided  his  property  between  his  two  sons. 
At  the  end  of  tlie  first  year  the  elder  had  spent  ono-quarter 
of  his,  and  the  younger  had  made  $1000,  and  their  i)roperty 
was  then  equal.  After  this  the  elder  spent  ?^500,  and  the 
younger  made  $2000,  when  it  appeared  that  the  younger  had 
just  double  the  elder;  what  had  each  from  the  father? 

J  Elder,         $4000. 
^^*  I  Younger,  $2000. 

15.  If  John  give  Charles  15  apples,  they  will  have  the 
tame  number;  but  if  Charles  give  15  to  John,  John  will 
have  15  times  as  many,  wanting  10,  as  Charles  will  have  left. 
How  many  hits  each  ?  a       \  John,       60. 


Ans.  \ 


Charles,  20. 

10.  Two  clerks,  A  and  J?,  have  salaries  which  are  together 
equal  to  $900.  A  spends  ^\  per  year  of  what  he  receives, 
and  B  adds  as  much  to  his  as  A  spends.  At  the  end  of  the 
year  they  have  equal  sums :  what  was  the  salary  of  each  ? 

.        S  A's  =  $500. 
^^^•Ii?',  =  $400. 

Elimination  by  Substitution, 

114.     Let  us  again  take  the  equations, 

5a!-f  Vy  =  43,  (1.) 

lla^^-  9y  =  C9.  (2.) 

1 14  H've  the  rule  for  eluniDation  by  suhfititutioii.  When  \b  thio  n><>thod 
wfA  to  the  greatmit  advanta^  T 


136  ELEMENTARY      ALGEBRA. 

Find  the  value  of  a;  in  the  first  equation,  which  gives, 

"^  -  5 

Substitute  this  value  of  a;  in  the  second  equation,  and  we 
have, 

43  —  7v 
11   X  5-^  +  92/  =  69; 

or,  473  —  11y  +  452/  =  345 ; 

or,  —  32y  =  —  128. 

Here,   a  has  been  eliminated  by  substitution. 

In  a  similar  manner,  w^e  can  eliminate  any  unknown  quan- 
tity ;  hence,  the 

RULE. 

I.  Find  from  either  equation  the  value  of  the  unJaiown 
quantity  to  be  eliminated  : 

n.  Substitute  this  value  for  that  quantity  in  the  other 
equation. 

Note.  —This  method  of  elimination  is  used  to  great  advan- 
tage when  the  coefficient  of  either  of  the  unknown  quantities 
is  1. 

EXAMPLES. 

Find,  by  the  last  method,  the  values  of  x  and  y  in  the 
following  equations: 

1.  3a;  —  2/  =   1,    and    3y  —  2a5  =   4. 

Ans.   a;  =  1,    2/  =  2 

2.  5y  —  4a;  =   —  22,    and    Sy  -^  Ax  —  38. 

Alls,   a;  =  8,    y  =  2. 

3.  a;  +  81/   =  18,    and    y  —  Zx  —   —  29. 

Ans.   a;  =   10,    y  =   L 


ELIMINATION.  137 


2 

4.    bx  -  y  =   13,    and    8a;  +  -y  =  29. 


Ans.   05  =   3^,    y  r=  4i 


6.    10a;  ~  I  =   09,    and    lOy  -  ^  =  49. 
5  7 


Ans.   a  =  7,    y  =  5. 


6.    «  +  |»  -  f  =  10,    and    I  +  ^  =  2. 


Ans.   a;  =  8,    y  =  10. 


».  1-1  +  5  =  2,  «  +  !  =  in. 


^w«.  a;  =  15,    y  =  14. 


1  +  1  +  3  =  6^    ana    1-1  =  1. 

^?i5.   a;  =  3^,    y  =  4. 


^/i5.  aj  =  12,    y  =  16. 

10.  ?  -  ?  -  1   =   -  9,    and    5a;  ~  ^  =  29. 
7         2  '  49 

Ans.  a;  =  6,    y  =  7. 

11.  Two  misers,  -4  and  -S,  sit  douTi  to  count  over  their 

money.     They  both  have  $20000,  and  3  has  three  times  aa 

much  as  A  :  how  much  has  each  ?  ,   ^     a-^^^ 

.         J  -4,    loOOO. 

^''^'    ]  J?,  $15000. 

12.  A  person  has  two  purses.  If  he  puts  $7  into  the  first, 
the  whole  is  worth  three  times  as  much  as  the  second  purye: 
but  if  he  puts  $7  into  the  second,  the  whole  is  worth  liv^e 
times  as  much  as  the  first :  '^hat  is  the  value  of  each  purse  ? 

Ans.    iBt,  $2;  ?d,  $3. 


138  ELEMEiq^TAEY      ALGEBEA. 

13.  Two  numbers  have  the  following  relations:  if  the 
first  be  multiplied  by  6,  the  product  will  be  equal  to  the 
second  multiplied  by  5 ;  and  1  subtracted  from  the  first 
leaves  the  same  remainder  as  2  subtracted  from  the  second : 
what  are  the  numbers  ?  Ans.  5  and  6. 

14.  Find  two  numbers  with  the  following  relations:  the 
drst  increased  by  2  is  3J  times  as  great  as  the  second  *, 
and  the  second  increased  by  4  gives  a  number  equal  to  half 
the  first:  what  are  the  numbers?  A7is.  24  and  8. 

15.  A  father  says  to  his  son,  "twelve  years  ago,  I  was 
twice  as  old  as  you  are  now:  four  tunes  your  age  at  tliat 
time,  plus  twelve  years,  will  express  my  age  twelve  years 
hence  ; "    what  w^ere  their  ages  ? 


~o^ 


.         (  Father,  12  years. 
^''^-    (Son,        30       " 


^Elimination  hy  Comijarison. 

115.     Take  the  same  equations, 

5iB  +  7y  =  43 
11a;  +  92/  =  69. 

Finding  the  value  of  x  from  the  first  equation,  we  have, 

43  —  Vy 

aj  :=   • « 

5        , 

and  finding  the  value  of  x  from  the  second,  we  obtain, 

69  —  9y 

X    =: —  • 

11 

115.  Give  the  rule  for  elimiuatiO'n  by  cotnpsu-isoiu 


ELIMINATION.  139 

Let  these  two  values  of  x  be  placed  equal  to  each  other, 
and  we  have, 

43  —  It/  _  09  —  9y 
5         "■         11 

Or,  473  —  ny  =  345  —  45y; 

or,  —  32y  =   —  128. 

llenco,  y  =  4. 

A    1  69  —  30         „ 

And,  X  =  — — - —  =  3. 

This  method  of  elimination  is  called  tlie  method  by  com- 
fcarison,  for  which  we  have  the  following 

•RULE. 

L  Find^  from  each  equation^   the  value  of  the  same 
unknown  quantity  to  he  eliminated': 
II.  Place  these  values  equal  to  each  other, 

EXAMPLES. 

Find,  by  the  last  rule,  the  values  of  x  and  y,  from  the 
follo\i'ing  equations, 

1.    3a;  +  I  +  0   =  42,    and    y  _  ^   =    14^. 

Ans.   a;  =   11,    y  =   15. 


2. 

l-f  +  5.6,    and    1  +  4  =  ^+6. 

Ans.  ar  =  £8,    y  =  20. 

3. 

^  -  f  +  f  =   1,    and    3y  -  »=  =  6. 

Ans.  a;  =  9,    y  =  6. 

4. 

y  -  3  =  „a;    f-  5     and    ^-^^  =  y  -  3^  • 

Ans.   a;  =  2,    y  =  9 

140  ELEMENTARY      ALGEBKA. 

5.    ^— ~ h  -  =  y  -  2,    and    -  +  ?  =  a  -  13. 

3  2         ^  '  8        7 

Ans.   aj  =  16,    y  =  1 

.       6.    ^^-5^ h  ^ =  05 ^ ,    and    a;  4-  y  =  16. 

^725.   a;  =   10,    y  =  6. 


/ 


2 

- 

0. 

Ans. 

a; 

=   1, 

y  = 

:    3. 

cc  -  4 
3 

= 

y  - 

X 

'  5* 

t/i5.  ar 

= 

10, 

2/  = 

13. 

^     2aj  —  3y 

1.    --^  =  a  -  2§,    a-  - 


8.    2y  4-  3a;  =  y  4-  43,    ?/  - 


9.    4y  -  ^-^-^  =  aj  +  18,   ^nd   27  -  y  =  a;  +  y  +  4. 

Ans.   a;  =  9,    y  =  1. 

10.     l_^  +  4   =  y-16|,    |-2   =  |. 

-4w5.   a;  =  10,    y  =  20. 


116  Having  explained  the  principal  methods  of  elimina- 
tion,  wo  shall  add  a  feAv  examples  which  may  be  solved  by 
any  one  of  them  ;  and  often  indeed,  it  may  be  advantageous 
to  emj)loy  them  all,  even  in  the  same  example. 

GENERAL      EXAMPLES. 

Find  the  values  of  x  and  y  in  the  following  simultaneous 
equations : 

1.    2a;  4-  3y  =  16,    and    3a;  —  2y  =  11. 

Ans.  a;  =  5,    y  =•     2. 


ELIMINATION, 


1« 


2,      L  _ii  =  _-       and    ■ — ~  z=.  — 

ft   ^   4  20*  4^5  120 


A  1  1 


3.  ?  +  7y  =  99,    and    ^  +  1x  =  51. 

Ans.  25  ==  7,    y  =  14 

4.  1-12=1  +  8.    ^l  +  l-S  =  'l^  +  .1. 


6. 


4o* 
*  -  ly  +  y   =   C} 


uln«.  a;  =  60,    y  =  40. 
a;  =  6. 


---  +  7.  =  41 

ia;  -  2^  +  4iy  =  12}      j 
5 


3y»~  g  ,   2a;  -  y 
6        "^        4 


6a;-y +  ?-^  =  ^^^  j 
'3a;  -  8       y  —  6 


Ans. 


Ans. 


Ans, 


+  "—T-  +  y  =  18 


TT 


8.^ 


9. 


8a  -  3  — 


6  -  y 


79 


4j;  —  4       V  —  5 


ix-iy  + 


V-* 


Ans. 


Ans. 


y  =  8 

a;  =  6. 

y  =  s. 

a;  =  9. 

y=8. 

X  =  10. 

y=  12 
x=:e. 

y  =  5. 


\/ 


/ 


142 


10. 


ELEMENTARY      ALGEIJllA. 

c  -\-  ab  —  bd 
X  = 

A71S. 


ax    —    by  =  c 
a  —  y  -^r  X  =  d 


a  —  b 


y  z= 


a^  +  c  —  ad 
a  —  b 


J  J    j  IS-T  +  Ty- 341  :=.  71^+4313;)  \x=-\2 

12    i(«^  +  5)(y+7)  =  (a;+l)(y-9)  +  112)  (  aj  = 

•   (2a:+10-32/-hl  f  ^''^- (  2/ ^ 


13.  < 


14.  ^ 


15. 


IG.  < 


ax  —  by 

x-^  y  —  c 

ax  -i-  by  =  c 
fx  +yy  =  h 


Ans. 


x  = 


Ans. 


X  = 


y  = 


=  3. 

y  —  5. 

be 
a^-b 
__     ac 
~  a-\-b' 

eg  —  bh 

ah  —  ef 
ag-bf^ 


Ans. 


in,  < 


b  +  y        Sa  -\-  X 
ax  +  2by  =  d 

bex  z=  cy  —  ^ 


,,     ,  <c'  -  b')       2b^  ,     , 
oc  e 


(sb  -  2f)bn 

'         ^  ^2  f2 


X   ~ 


y 


3a 

3a2  _  ^2  ^  ^ 

36 


y  -  a  = 


b'^-P 


Ans. 


Ans. 


|^  =  £ 


c 


y  = 


X  = 


PROBLEMS.  143 

PR0BLE3IS. 

1.  What  fi-action  is  that,  to  the  numerator  oi  which  if  1 

he  added,  the  value  will  be  - ,  V/Ut  if  1  be  added   to  its 

^1 
denominator,  the  value  will  be    -  ? 

X 

Let  the  fraction  be  denoted  by  -  • 

^  y 

Then,  by  the  conditions, 

SB  +  1         1  1         aJ  1 

-y-  =  5'  '"'^'  WTx  =  V 

whence,       3x  +  3  =  y,    and    ix  =  y  +1. 
Therefore,  by  subtractuig, 

X  —3  =  1,    and    aj  =  4. 
Hence,  12  +  3  =  y; 

.-.    y  =  15. 

2.  A  market-woman  bought  a  certain  number  of  ogg!^  at 
2  for  a  penny,  and  as  many  others  at  3  for  a  penny ;  and 
having  sold  tliciu  all  togetlier,  at  th<3  rate  of  5  for  2cl,  found 
that  she  had  lost  4d:  how  many  of  both  kinds  did  she  buy  ? 

Let  2x     denote  the  whole  number  of  eggs. 

Then,  x  =     the  number  of  eggs  of  each  sort. 

Then  will,     -x  =     the  cost  of  the  first  sort, 

and,  -X  =     the  cost  of  the  second  sort. 

3 

But,  by  the  conditions  o^  the  question, 

4x 
5:2«::2:_; 

Ax 
hence,    — -    will  denote  the  amount  for  which  the  eggs 

were  sold. 


144  ELEMENTARY      ALOE  UK  A. 

But,  by  the  conditions, 

therefore,  15a;  +  lOaj  —  24a;  =  120; 

,'.     a;  =  120  ;  the  number  of  eggs  of  each  sort. 

3.  A  person  possessed  a  capital  of  30,000  dollars,  for 
which  he  received  a  certain  interest ;  but  he  owed  the  sum 
of  20,000  dollars,  for  which  he  paid  a  certain  annual  interest. 
The  interest  that  he  received  exceeded  that  which  he  paid 
by  800  dollars.  Another  person  possessed  35,000  dollars,  for 
which  he  received  interest  at  the  second  of  the  above  rates ; 
but  he  owed  24,000  dollars,  for  w^hich  he  paid  interest  at  the 
first  of  the  above  rates.  The  interest  that  he  received,  an- 
nually, exceeded  that  w^hich  he  paid,  by  310  dollars.  Re- 
quu'ed  the  two  rates  of  interest. 

Let  X  denote  the  number  of  units  in  the  first  rate  of 
interest,  and  y  the  unit  in  the  second  rate.  Then  each  may 
be  regarded  as  denoting  the  interest  on  $100  for  1  year. 

To  obtam  the  interest  of  $30,000  at  the  first  rate,  denoted 
by  aj,   we  form  the  proportion, 

30,000a; 
100  :  30,000  : :  a;  :      '         ,  or  300a;. 

And  for  the  interest  of  $20,000,  the  rate  being    y, 

100  :  20,000  : :  2/  :  ^^y^j  or  200y. 

But,  by  the  conditions,  the  difference  between  these  two 
amomits  is  equal  to  800  dollars. 

We  have,  then,  for  the  first  equation  of  the  problem, 

300a;  —  200y  =  800 


PKOULEMS.  145 

By  expres^^  algebraically,  the  second  condition  of  the 
problem,  we  oi!Bn  a  second  equation, 

)y  —  240a;  =  310. 

Both  members  of  ^J|rst  equation  being  divisible  by  100 
and  those  of  the*%*^oiK^^10,  we  have, 

^  ^^ 

If  SiB  —  22,   =  8,         35y  —  24a;  =  31. 

To  eliminate  a;,  multiply  the  first  equation  by  8,  and  then 
add  the  result  to  the  second ;  there  results, 

19y  =  95,     whence,     y  =  6. 

Substituting  for  y,  in  the  first  equation,  this  value,  and 
that  equation  becomes, 

3a;  —  10  =  8,    whence,    a  =  6. 

Therefore,  the  first  rate  is  6  per  cent,  and  the  second  6. 

VERIFICATION. 

$30,000,     at  6  per  cent,  gives    30,000  X  .06  =  $1800. 
$20,000,  6  "  "         20,000  X  .05   =  $1000. 

And  we  have,       1800  —  1000  =  800. 

The  second  condition  can  be  verified  in  the  same  manner. 

4.  What  two  numbers  are  those,  whose  difference  is  Y, 
and  sum  33  ?  Ans,  13  and  20. 

5.  Divide  the  number  15  into  two  such  parts,  that  three 
times  the  greater  may  exceed  seven  times  the  less  by  15. 

Ans,  54  and  21. 

6.  In  a  mixture  of  wine  and  cider,  \  of  the  whole  plus  25 
gallons  was  wine,  and  ^  part  minus  5  gallons  was  cider :  how 
many  gallons  were  there  of  each  ? 

Ans.  85  of  wine,  and  36  of  cider. 
1 


146  ELEMENTARY       ALGEBRA. 

V.  A  bill  of  £120  was  paid  in  guineas  and  moidoies,  and 
the  number  of  pieces  used,  of  both  sorts,  was  just  100.  If 
the  guinea  be  estimated  at  21s,  and  the  moidore  at  27s,  how 
many  pieces  were  there  of  each  sort  ?  Ans.  60. 

8.  Two  travelers  set  out  at  the  same  time  from  London 
and  York,  whose  distance  apart  is  150  miles.  One  of  thera 
travels  8  miles  a  day,  and  the  other  7 :  in  what  time  wiU 
they  meet  ?  Ans.  In  10  days. 

9.  At  a  certain  election,  375  persons  voted  for  two  candi- 
dates, and  the  candidate  chosen  had  a  majority  of  91  :  how 
many  voted  for  each  ? 

A71S,  233  for  one,  and  142  for  the  other. 

10.  A  person  has  two  horses,  and  a  saddle  worth  £60. 
Now,  if  the  saddle  be  put  on  the  back  of  the  first  horse,  it 
makes  their  joint  value  double  that  of  the  second  horse; 
but  if  it  be  put  on  the  back  of  the  second,  it  makes  their 
joint  value  triple  that  of  the  first :  what  is  the  value  of  each 
horse  ?  Ans,  One  £30,  and  the  other  £40. 

11.  The  hour  and  minute  hands  of  a  clock  are  exactly  to- 
gether at  12  o'clock :  when  will  they  be  again  together? 

Ans.  Ih.  5y\m« 

12.  A  man  and  his  wife  usually  drank  out  a  cask  of  beer 
in  12  days ;  but  when  the  man  was  from  home,  it  lasted  the 
woman  30  days :  how  many  days  would  the  man  alone  be 
in  drinking  it  ?  Ans.  20  days. 

•  13.  If  32  poimds  of  sea-water  contain  1  pound  of  salt,  how 
much  fresh  water  must  be  added  to  these  32  pounds,  in  order 
that  the  quantity  of  salt  contained  in  32  pounds  of  the  new 
mixture  shall  be  reduced  to  2  ounces,  or  |  of  a  pound  ? 
'-  /^2/'^)  -=-    ^.'  ■^^^^-  224  lbs. 

T4.  A  person  who  possessed  100,000  dollars,  placed  the 
greater  part  of  it  out  at  5  per  cent  interest,  and  the  otlier 


PE0BLEM8.  147 

at  4  per  cent.   The  interest  which  he  recei\  ed  for  the  whole, 
amounted  to  4640  dollars.     Required  the  two  parts. 

Ans.    164,000  and  $30,000. 

15.  At  the  close  of  an  election,  the  successful  candidate 
had  a  majority  of  1500  votes.  Had  a  fourth  of  the  votes  of 
the  unsuccessful  candidate  been  also  given  to  him,  he  would 
have  received  three  times  as  many  as  his  competitor,  want- 
hig  three  thousand  five  hundred :  how  many  votes  did  each 
receive?  .         j  1st,  6500. 

I  2d,  5000. 

16.  A  gentleman  bought  a  gold  and  a  silver  watch,  and  a 
chiun  worth  |25.  When  he  put  the  chain  on  the  gold  watch> 
it  and  the  chain  became  worth  three  and  a  half  times  more 
than  the  silver  watch ;  but  when  he  put  the  chain  on  the 
silver  watch,  they  became  worth  one-half  the  gold  watch 
and  15  dollars  over :  what  was  the  value  of  each  watch  ? 

J         j  Gold  watch,  $80. 
^"^^    (Silver    «      $30. 

17.  There  is  a  cert^  number  expressed  by  two  figures, 
which  figures  are  called  digits.  The  sum  of  the  digits  is  11, 
and  if  13  be  added  to  the  first  digit  the  sum  will  be  three 
times  the  second:  what  is  the  nimiber?  Ans.  56. 

18.  From  a  company  of  ladies  and  gentlemen'  16  ladies 
retire;  there  are  then  left  two  gentlemen  to  each  lady. 
After  which  45  gentlemen  depart,  when  there  are  left  5 
ladies  to  each  gentleman :  how  many  were  there  of  each  at 
first  ?  J         j  ^^  gentlemen. 

(  40  ladies. 

1 9.  A  person  wishes  to  dispose  of  his  horse  by  lottery. 
If  he  seUs  the  tickets  at  $2  each,  he  will  lose  $30  on  his 
borpe;  but  if  he  soils  them  at  $3  each,  he  wall  receive  $30 


148  ELEMENTARY      ALGEBKA. 

more  than  his  horse  cost  him.     What  is  the  value  of  the. 

horse,  and  number  of  tickets?        .         (Horse,  $150. 

Ans.   <  -^       '      , 

(  No.  of  tickets,  60, 

20.  A  person  purchases  a  lot  ot  wheat  at  $1,  and  a  lot  of 
rye  at  75  cents  per  bushel ;  the  whole  costing  him  $117.50. 
He  then  sells  |  of  his  wheat  and  ^  of  his  rye  at  the  same  rate, 
and  realizes  $27.50.     How  much  did  he  buy  of  each? 

.         (  80  bush,  of  wheat. 


-1 


50  bush,  of  rye. 

21.  There  are  52  pieces  of  money  in  each  of  two  bags.  A 
takes  from  one,  and  JB  from  the  other.  A  takes  twice  as 
much  as  -S  left,  and  J^  takes  7  times  as  much  as  A  left. 
How  much  did  each  take?  .         j  ^,  48  pieces. 


[A, 


28  pieces. 

22.  Two  persons,  A  and  J?,  purchase  a  house  together, 
worth  $1200.  Says  A  to  -B,  give  me  two-thirds  of  your 
money  and  I  can  purchase  it  alone ;  but,  says  ^  to  A^  if 
you  will  give  me  three-fourths  of  your  money  I  shall  be  able 
to  purchase  it  alone.     How  much  had  each  ? 

A71S.   A,  $800 ;  -S, 


23.  A  grocer  finds  that  if  he  mixes  sherry  and  brandy  in 
the  proportion  of  2  to  1,  the  mixture  will  be  worth  78s.  per 
dozen ;  but  if  he  mixes  them  in  the  proportion  of  7  to  2,  he 
can  get  795.  a  dozen.  What  is  the  price  of  each  liquor  per 
dozen?  Ans.   Sherry,  81s.;  brandy,  72s. 

Equations  containing  three  or  more  unknown  quantities 

117.     Let  us  noT\' consider  equations  involving  three  or 
more  unknown  quantities. 
Take  the  group  of  simultaneous  equations, 

117.  Give  the  rule  for  solving  any  group  of  simultaneous  equations? 


EX  AMTLES. 

1^ 

5a;  -  6y  +  42  =  16, 

.     .      (1.) 

7aj  4-  4y  -  Sz  =  19, 

.     .       (2.) 

2a;  +    2/  -h  63  =  46.      .     , 

.     .       (8.) 

149 


To  eliminate  2  by  means  of  tlie  first  two  equations,  multi- 
ply the  first  by  3,  and  the  second  by  4 ;  then,  since  the 
coeflicients  of  2  have  contrary  signs,  acjd  the  two  results 
together.    Tliis  gives  a  new  equation : 

43aj  -  22,   =  121 (4.) 

Multiplying  the  second  equation  by  2  (a  factor  of  the 
coefficient  of  2  in  the  third  equation),  and  adding  the  result 
to  the  third  equation,  we  have, 

16a;  +  9y  =  84 (5.) 

The  question  is  then  reduced  to  finding  the  values  of  x 
and  y,  which  will  satisfy  the  new  Equations  ( 4 )  and  ( 5 ). 

Now,  if  the  first  be  multii^lied  by  9,  the  second  by  2,  and 
the  results  added  together,  we  find, 

41905  =  1257;  whence,    a;  =  3. 

We  might,  by  means  of  Equations  (4)  and  (6)  deter- 
mine y  in  the  same  way  that  we  have  determined  x ;  but 
the  value  of  y  may  be  detennined  more  simply,  by  substi- 
tuting the  value  of  a;  in  Equation  ( 6  )  ;  thug, 

48  +  9y  =   84.  .  • .     y  =  ^^  ^  ^^   =  4. 

In  the  same  manner,  the  first  of  the  three  given  equations 
becomes,  by  substituting  the  values  of  x  and  y, 

15  —  24  -I-  42  =  15  .-.     z  =  ~  =  6, 

4 

In  the  same  way,  any  gronr»  cC  «imiiltantous  equations 
may  be  solved      Hence,  the 


150         ELEMENTAEF   ALGEBRA. 
KULE, 

1.  Combine  one  equation  of  the  group  with  each  of  the 
others,  by  eliminating  one  unknown  quantity ;  there  will 
result  a  new  group  containing  one  equation  less  than  the 
original  group: 

n.  Combine  one  equation  of  this  new  group  with  each 
of  the  others,  by  eliminatirig  a  second  unknown  quantity  ; 
there  will  result  a  new  group  containing  two  equations  less 
than  the  original  group : 

in.  Conti9iue  the  operation  until  a  shigle  equation  is 
found,  containing  but  one  unknown  quantity  : 

lY.  Find  the  value  of  this  unknown  quantity  by  the 
preceding  rules  y  substitute  this  in  one  of  the  group  of 
two  equations,  and  find  tlie  value  of  a  second  unknown 
quantity ;  substitute  these  in  either  of  the  group  of  three^ 
finding  a  third  unknown  quantity ;  and  so  on,  till  the 
values  of  all  are  found. 

Notes. — 1.  In  order  that  the  value  of  the  unknown  quan- 
(;ities  may  be  determined,  there  must  be  just  as  many  inde- 
pendent equations  of  condition  as  there  are  imknown  quan- 
tities. If  there  are  fewer  equations  than  unkno^vn  quantities, 
the  resulting  equation  will  contain  at  least  two  unknown 
quantities,  and  hence,  their  values  cannot  be  found  (Art.  110). 
If  there  are  more  equations  than  unknown  quantities,  the 
conditions  maybe  contradictory,  and  the  equations  impossible. 

2.  It  often  happens  that  each  of  the  proposed  equations 
does  not  contain  all  the  unknown  quantitAes.  In  this  case, 
with,  a  little  address,  the  elimination  is  very  quickly  per- 
formed. 

Take  the  four  equations  involving  four  unknown  quanti- 
ties: 

2a;  -  3y  +  22  =   13.     (1.)  4?/  +  23  =  14.     (3.) 

4w  -  2a;  =:=  30.     (2.)  6y  -f  ^u  =  32.     (4.) 


BXi^MPLES.  151 

By  inspecting  thase  equations,  we  see  that  the  elimination 
of  z  in  the  two  Equations,  (  1 )  and  (  3 ),  will  give  an  equar 
tion  involving  x  and  y;  and  if  we  eliminate  u  in  Equa- 
tions ( 2 )  and  ( 4 ),  we  shall  obtain  a  second  equation,  in- 
vohnng  x  and  y.  These  last  two  unknown  quantities  may 
therefore  be  easily  determined.  In  the  first  place,  the 
elimination  of  z  from  ( 1 )  and  (  3  )  gives, 

7y  —  2aj  =  1 ; 

That  of  u  from  (  2  )  and  ( 4  )  gives, 

20y  4-  6a;  =  38. 

Multiplying  the  first  of  these  equations  by  3,  and  adding, 
41y  =  41; 

Whence,  y  =     1. 

Substituting  this  value  in    Ty  —  2a;  =  1,    we  find, 

a;  =  3. 

Substituting  for  x  its  value  in  Equation  ( 2 ),  it  becomes 

4w  —  6  =  30. 

Whence,  w  =  9. 

And  substituting  for  y  its  value  in  Equation  (3),  there 
reeulte, 

z  =  6. 

EXAMPLES. 


«  +    y  H-    z  =  29 
SB  -f  2y  4-  32  =  62 


1.  Given  ^  .         ^         ^ 


^  +  ^  +  ^  =   10 

A718.  X  =  8,    y  =  9,    z  =  12, 


►  to  find  JB,  y,  and  e. 


162 


LEMENTARF        ALGEBRA. 


r  2a;  4-  4y  —  32  =  22   | 
2.  Given  <  4x  —  2y  -{-  5z  =  18   I  to  finrl  x,  y,  and  z. 
[  6a;  +  Vy  —    s  =  63  J 

Ans.  a;  =  3,    y  —  7,    s  =  4, 


3.  Given  < 


35  4-  2^  +  3^  =  32 

o^^  +  72/  +  ^2  =  15   r  *^  fi^^  ^'  y?  and  2 
o  4  5 

Ans.   a;  =   12,    y  =  20,    z  =  30. 


4.  Given 


r  a;  +  y  +  2  =  29i 
<  X  +  y  —  z  =  ISl 
I  a;  —  y  +  s  =   13| 


294-   , 

4^   ^  to  find  a;,  y,  and  z. 


Ans.   a;  =  16,    y  =  Yf,    s  =  5J 


3a;  +  5y  =  161 
5.  Given  -{   Ya;  +  22    =  209   }»  to  find  a;,  y,  and  z. 
2y  +    2    =     89 

-4715.   a;  r=  17,    y  =  22,    2  =  46. 


V 


ri     1 


6.  Given  ^ 


-  -f —  =   5   }-  to  find  a;,  y,  and  2. 


L  y       s  J 


a;  = 


2 


a  +  b  —  c' 


y 


a  -]-  G  —  b' 


z  = 


h-\-c  —  a 


Note. — In  this  example  we  should  not  proceed  to  clear 
the  equation  of  fractions;  but  subtract  immediately  the 
second  equation  from  the  first,  and  then  add  the  third  ;  we 
thus  find  the  value  of  y. 


PROBLEM  8.  153 


PROBLEMS. 

1.  Divide  the  number  90  into  four  such  parts,  that  the 
first  increased  by  2,  the  second  diminished  by  2,  the  third 
multiplied  by  2,  and  the  fourth  divided  by  2,  shall  be  equal 
each  to  each. 

This  problem  may  be  easily  solved  by  introducing  a  new 
unlaio\Ma  quantity. 

Let  X,  y,  2,  and  ?/,  denote  the  required  parts,  and  desig. 
nate  by  ni  the  several  equal  quantities  which  arise  from  the 
conditions.     We  shall  then  have, 

u 

X  +  2  z=z  m,     y  —  2  =  r«,     2z  =  m,     -   =  m. 


2 


X  =  m  —  2,    y  =  r7i4-2,     z  =  — ,     u  =  2m, 


From  which  we  find, 
X  =  m  —  2,    y  - 
And,  by  adding  the  equations. 


m 
SB-f-y  +  24-w  =  m  -\-  m  -\-  —  -\-  2m  =  4\m, 

Si 

And  since,  by  the  conditions  of  the  problem,  the  first 
member  is  equal  to  90,  we  have, 

\\m  =  90,     or    \m  =  90; 

hence,  m  =  20. 

Having  the  value  of  m,  we  easily  find  the  other  values; 
viz.: 

jc  =   18,     y  =   22,     z  =   10,     u  =  40. 

2.  There  are  three  ingots,  composed  of  different  metals 
mixed  together.  A  pound  of  the  first  contains  7  ounces  of 
silver,  3  ounces  of  copper,  and  6  of  pewter.  A  pound  of 
the  second  contains  12  ounces  of  silver,  3  ounces  of  copj)er, 
and  1  of  pewter.  A  pound  of  the  third  contains  4  ounces 
of  silver,  7  ounces  of  copper,  and  5  of  pewt.or.  It  is  required 
7* 


154  ELEMENTARY      ALGEBRA. 

to  find  how  much  it  will  take  of  each  of  the  three  ingots  to 
form  a  fom-th,  which  shall  contain  in  a  pound,  8  ounces  of 
silver,  3|  of  copper,  and  4^  of  pewter. 

Let  jc,  2/,  and  ^,  denote  the  number  of  ounces  which  it 
is  necessary  to  take  from  the  three  ingots  respectively,  in 
order  to  form  a  pound  of  the  required  ingot.  Since  there 
are  V  ounces  of  silver  in  a  pound,  or  16  ounces,  of  the  first 
ingot,  it  follows  that  one  ounce  of  it  contains  ^^  of  an  ounce 
of  silver,  and,  consequently,  in  a  number  of  ounces  denoted 

by  (K,  there  is  —  ounces  of  silver.     In  the  same  manner, 

1 2ii/  4  z 

we  find  that,   -^ ,  and  — ,  denote  the  number  of  ounces 

of  silver  taken  from  the  second  and  third ;  but,  from  the 
enunciation,  one  pound  of  the  fourth  ingot  contains  8  ounces 
of  silver.     We  have,  then,  for  the  first  equation, 

!^  4.  l?^  -J-  1^    ::^    8  • 
16  "^    16         16  ' 

or,  clearmg  fractions^ 

1x  +  12y  +  42  =   128. 

As  respects  the  copper,  we  should  find, 

305  4-  3y  4-  T2  =  60 ; 

and  with  reference  to  the  pevki;er, 

Qx  +  y  +  5z  =  68. 

As  the  coefficients  of  y  in  these  three  equations  are  the 
most  simple,  it  is  convenient  to  eliminate  this  unknown 
quantity  first. 

Multiplying  the  second  equation  by  4,  and  subtracting  the 
first  from  it,  member  from  member,  we  have, 

5x  +  242   =    112. 


PB0BLEM6.  155 

Multiplying  the  third  equation  hj  3,  and  subtracting  the 
second  from  the  resulting  equation,  we  have, 

1525  +  83  =  144. 

Multiplying  this  last  equation  by  3,  and  subtracting  the 
preceding  one,  we  obtain, 

40a;  =  320; 
whence,  a;  =  8. 

Substitute  this  value  for  x  in  the  equation, 
16a;  -f-  82  =  144; 
it  becomes,  120  +  82  =  144, 

whence,  2  =  3. 

Lastly,  the  two  values,  a;  =  8,  2  =  3,  bemg  substituted 
in  the  equation, 

6aj  +  y  -h  52  =  68, 
give,  48  +  y  +  15  =  68, 

whence,  2/  =     ^* 

Therefore,  in  order  to  form  a  pound  of  the  fourth  ingot, 
we  must  take  8  ounces  of  the  first,  6  ounces  of  the  second, 
and  3  of  the  third. 

VEEIFICATION. 

K  there  be  1  ounces  of  silver  in  16  ounces  of  the  first 
ingot,  in  eight  ounces  of  it  there  should  be  a  number  of 
ounces  of  silver  expressed  by 

7x8 

16 
In  like  manner, 

12  X  5  ,4x8 

,   and    , 

16      '  16     ' 

will  express  the  quantity  of  silver  contamed  in  5  ounces  of 
the  second  ingot,  and  3  ounces  of  the  third. 


156  ELEMENTARY      ALGEBRA. 

Now,  we  have, 

7X8        12   X  5        4x3    _    128   _   g. 
~1l6        ^       16        "^       16       ""     16     ""      ' 

therefore,  a  pound  of  the  fourth  ingot  contains  8  ounces  of 
silver,  as  required  by  the  enunciation.  The  same  conditions 
niay  be  verified  with  respect  to  the  copper  and  pewter. 

3.  A''s  age  is  double  Ij\%  and  ^'s  is  triple  of  C»5,  and  the 
sum  of  aU  their  ages  is  140  :  what  is  the  age  of  each  ? 

Ans.   A's  —  84;  B's  =  42;  and  G's  =  14. 

4.  A  person  bought  a  chaise,  horse,  and  harness,  for  £80 ; 
the  horse  came  to  twice  the  price  of  the  harness,  and  the 
chaise  to  twice  the  cost  of  the  horse  and  harness :  what  did 
he  give  for  each?  (  £13    6s.  8c?.  for  the  horse. 

Ans.  \    £6  13s.  4f?.  for  the  harness. 
(  £40  for  the  chaise. 

5.  Divide  the  number  36  into  three  such  parts  that  \  of 
the  first,  \  of  the  second,  and  ^  of  the  third,  may  be  all 
equal  to  each  other.  Aiis.  8,  12,  and  16. 

6.  If  A  and  B  together  can  do  a  piece  of  work  in  8  days, 
A  and  C  together  in  9  days,  and  B  and  C  in  ten  days,  how 
many  days  would  it  take  each  to  perform  the  same  work 
alone?  Ans.   A,  14ff;  J5,  l^H;   G,  S^/y. 

7.  Three  persons,  A^  J5,  and  C,  begin  to  play  together, 
having  among  them  all  $600.  At  the  end  of  the  first  game 
A  has  won  one-half  of  J5's  money,  which,  added  to  his  own, 
makes  double  the  amount  B  had  at  first.  In  the  second 
game,  A  loses  and  B  wins  just  as  much  as  G  had  at  the  be- 
ginning, when  A  leaves  off  with  exactly  Avhat  he  had  at  first : 
now  much  had  each  at  the  beginning  ? 

Ans.   ^,$300;  i?,  $200 ;   G  $100. 

8.  Three  persons,  A,  B,  and  (7,  together  posscs.«  $3640. 


PE0BLEM6.  157 

It*  B  gives  A  $400  of  his  money,  then  A  will  have  $320 
more  than  B\  but  if  B  takes  $140  of  C''s  money,  then  B 
and  C  will  have  equal  sums  :  how  much  has  each  ? 

Alls,   A,  $800  ;,  B,  $1280 ;   C,  $1560. 

9.  Three  persons  have  a  bill  to  pay,  which  neither  alone 
is  able  to  discbarge.  A  says  to  J5,  "  Give  me  the  4th  of 
yonr  money,  and  then  I  can  pay  the  bill."  B  says  to  C 
•'Give  me  the  8th  of  yours,  and  I  can  pay  it."  But  C says 
to  A^  "  You  must  give  me  the  half  of  yours  before  I  can 
pay  it,  as  I  have  but  $8  "  :  what  was  the  amount  of  their 
bill,  and  how  much  money  had  A  and  B  ? 

.       j  Amount  of  the  bill,  $13, 
•  1^1  had  $10,    and  i?  $12. 

10.  A  person  possessed  a  certain  capital,  which  he  placed 
out  at  a  certain  interest.  Another  person,  who  possessed 
10000  dollars  more  than  the  first,  and  who  put  out  his  capital 
1  per  cent,  more  advantageously,  had  an  annual  income 
greater  by  800  dollars.  A  third  person,  who  possessed 
15000  dollars  more  than  the  first,  putting  out  his  capital  2 
per  cent,  more  advantageously,  had  an  annual  income  greater 
by  1500  dollars.  Required,  the  capitals  of  the  three  per- 
sons, and  the  rates  of  interest. 

.       j  Sums  at  interest,   $30000,  $40000,  $45000. 
*  ( Rates  of  interest,        4  6       '       6  pr.  ct. 

11.  A  widow  receives  an  estate  of  $15000  from  her  de- 
ceased  husband,  with  directions  to  divide  it  among  two  sons 
and  three  daughters,  so  that  each  son  may  receive  twice  as 
much  as  each  daughter,  and  she  herself  to  receive  $1000 
more  than  all  the  children  together :  what  was  her  share, 
and  what  the  «liare  of  each  child? 

i  The  widow's  share,  $8000 

Ans.  \  Each  son's,  $2000 

(  Each  daughter's,       $1000 


158  ELEMENTAKY      ALGEBRA. 

12.  A  certain  sum  of  money  is  to  be  divided  between 
three  persons,  A,  B^  and  G,  A  is  to  receive  |3000  less 
than  half  of  it,  B  $1000  less  than  one-third  part,  and  G  to 
receive  $800  more  than  the  fourth  part  of  the  whole  :  what 
is  the  sum  to  be  divided,  and  what  does  each  receive  ? 

Sum,  $38400. 


Ans» 


A  receives  $16200 
B  "  $11800. 
G        "        $10400. 


13.  A  person  has  three  horses,  and  a  saddle  which  is  worth 
$220.  If  the  saddle  be  put  on  the  back  of  the  first  horse,  it 
will  make  his  value  equal  to  that  of  the  second  and  third ; 
if  it  be  put  on  the  back  of  the  second,  it  will  make  his  value 
double  that  of  the  first  and  third ;  if  it  be  put  on  the  back 
of  the  third,  it  will  make  his  value  triple  that  of  the  first 
and  second :  what  is  the  value  of  each  horse  ? 

Ans,   1st,  $20;    2d,  $100  ;    3d,  $140. 

14.  The  crew  of  a  ship  consisted  of  her  complement  of 
sailors,  and  a  number  of  soldiers.  There  were  22  sailors  to 
every  three  guns,  and  10  over ;  also,  the  whole  number  of 
hands  was  five  times  the  number  of  soldiers  and  guns  to- 
gether. But  after  an  engagement,  in  which  the  slain  were 
one-fourth  of  the  survivors,  there  wanted  5  men  to  make 
13  men  to  every  two  guns:  requked,  the  number  of  guns, 
soldiers  and  sailors. 

Ans.  90  guns,  65  soldiers,  and  670  sailors, 

15.  Three  persons  have  $96,  which  they  wish  to  divide 
equally  between  them.  In  order  to  do  this,  A^  who  has  the 
most,  gives  to  B  and  G  as  much  as  they  have  already ;  then 
B  divides  with  A  and  G  in  the  same  manner,  that  is,  by 
giving  to  each  as  much  as  he  had  after  A  had  divided  with 
them  •   G  then  makes  a  division  with  A  and  i?,  when  it  Is 


PB0BLEM8.  159 

found  that  they  all  have  equal  sums :  how  m ach  had  each 
at  first?  Ans.  1st,  $52;    2d,  |28;    3d,  $16. 

16.  Divide  the  number  a  into  three  such  parts,  that  the 
first  shall  be  to  the  second  as  m  to  w,  and  the  second  to  the 
third  as  pto  q. 

^         amp  _         anp  _  a?iq 

~~  mp-{-np-\-nq^    ^  ~  mp-\-np-\-nq^       ~  mp-{-np-\-nq 

17.  Three  masons,  -4,  i?,  and  (7,  are  to  build  a  wall.  A 
and  B  together  can  do  it  in  12  days ;  B  and  C  in  20  days ; 
and  A  and  (7  in  15  days:  in  what  time  can  tach  do  it  alone, 
and  in  what  time  can  they  all  do  it  if  they  work  together  ? 

Ana.  Ay  in  20  days;  -B,  in  30 ;  and  C,  in  00 ;  all,  in  10, 


160  ELEMENTARY       ALGEBRA.. 


CHAPTER    VI. 


FORMATION   OP   POWERS 


118.  A  Power  of  a  quantity  is  the  product  obtained  by 
taking  that  quantity  any  number  of  times  as  a  factor. 

If  the  quantity  be  taken  once  as  a  factor,  we  have  the  first 
power ;  if  taken  twice,  we  have  the  second  power ;  if  three 
times,  the  third  power;  if  n  times,  the  n*^  power,  n  being 
any  whole  number  whatever. 

A  power  is  indicated  by  means  of  the  exponential  sign  ' 
thus, 

a  ^  a}   denotes  first  power  of  a.* 


ax  a  =  d?- 

(( 

square,  or  2d  power  of  a. 

ax  ax  a  =  a^ 

(( 

cube,  or  third  power  of  a. 

axaxaxa  =  a* 

(( 

fourth  power  of  a. 

axaxaxaxa  =  «® 

(( 

fifth  power  of  a. 

axaxaxa —  =  a"^ 

(( 

m*^  power  of  a. 

In  every  power  there  are  three  things  to  be  considered : 

1st.  The  quantity  which  enters  as  a  factor,  and  which  is 
called  the  first  power. 

2d.  Tlie  small  figure  which  is  placed  at  the  right,  and 
a  Uttle   above  the  letter,   is   called  the    exponent  of  the 

*  Since  a«  -  1  (Art.  49),  a"  X  o  =  1  X  a  =  a^ ;  so  that  the  two 
factors  of  a\  are  1  and  a. 

118.  What  is  a  power  of  a  quantity?  What  is  the  power  when  tte 
quantity  is  taken  once  as  a  factor  ?  When  taken  twice  ?  Three  times  f 
fi  times?  How  is  a  power  indicated  ?  In  every  power,  how  miny  things 
are  considered  ?    Name  them. 


POWKBS      OF      MONOMIALS.  161 

power,  and  shows  how  many  times  the  letter  enters  as  a 
fiictor. 

3d.  The  power  itself,  which  is  the  final  product,  or  result 
of  the  multiplications. 

POWERS     OF     l»IONO>nALS. 

119,  Let  it  be  required  to  raise  the  monomial  2a'5'  to 
the  fourth  power.    We  have, 

(2a^b^y  =  2a^b^  X  2a^h^  X  '2a^b^  X  2a?h\ 

which  merely  expresses  that  the  fourth  power  is  equal  to 
the  product  which  arises  from  taking  the  quantity  four 
times  as  a  factor.  By  the  rules  for  multiplication,  this  pro- 
duct is 

from  which  we  see, 

l8t.  That  the  coefficient  2  must  be  raised  to  the  4th 
power;  and, 

2d.  That  the  exponent  of  each  letter  must  be  multiplied 
hy  4,  the  exponent  of  the  power. 

As  the  same  reasoning  applies  to  every  example,  we  have, 
for  the  raising  of  monomials  to  any  power,  the  followmg 

RULE. 

I.  Raise  the  coefficient  to  the  required  power : 
IT.  Multiply  the  exponent  of  each  letter  hy  the  exponent 
of  the  power, 

EXAMPLES. 

1.  What  is  the  square  of  3ay  ?  Afis.   9a*i/^ 

110.  What  is  the  rule  for  raising  a  monomial  to  any  power?  WLen 
the  monomial  is  pogitive,  what  will  be  the  sign  of  its  powers  ?  When 
negative,  what  powors  will  be  plitf'  what  miuua? 


162  ELEMENTARY      ALGEBRA. 

2.  What  is  the  cube  of  Qa^y'^x'i  Ans,    21Qa^^y^x^, 

3.  What  is  the  fourth  power  of  2a^y^^?  16ai2yi2j2o^ 

4.  WTiat  is  the  square  of  a^js^s  p  Ans.   a^b^^y^. 


5 

6. 
1, 

What  is  the  seventh  power  of  a^hcd^  ? 

Mns.  a^^b'^c'd^^ 
What  is  the  sixth  power  of  a^b^c^d'i 

Ans.   a^^b'^cH^ 

What  is  the  square  and  cube  of    —  2a^'^? 

/Square.                                     Cube. 

—  2a^b^                                    —  2a252 

—  2a'^b^                                    -  2a^b^ 

+  4a^b\                                  -f  4a*5* 

-  2a2J2 

By  observing  the  way  in  which  the  powers  are  formed, 
we  may  conclude, 

1st.    When  the  monomial  is  positive,  all  the  powers  will 
be  2^ositive. 

2d.    When  the  monoTnial  is  negative,  all  even  poicers  loill 
be  positive,  and  all  odd  will  be  negative. 

8.  What  is  the  square  of  —  2^*5^  ?  Ans.   4:a^b^\ 

9.  What  is  the  cube  of  -^  da^'b^  ?       Ans.    —  125a^"b\ 

10.  What  is  the  eighth  power  of  —a^xy^  ? 

Ans.    -f  a^^x^y'^ 

11.  What  is  the  seventh  power  of  —  a»»J"c  ?. 

Ans.    —  a'''^b''^c^ 

12.  What  is  the  sixth  power  of  2a¥y^  ? 

Ans.    Ua^b'^^y^^, 


POWERS      OF      FRACTIONS.  163 

18.  What  is  toe  ninth  power  of  —  a^hc^  ? 

Ans.    —  a^^b^c^\ 

14.  What  is  the  sixth  power  of  —  Sab^d? 

Ans,   I29a^b^^d^ 

15.  What  is  the  square  of  —  10a'"5"c3  ? 

Ans.    lOOa^-'J^'ce, 

16.  What  is  the  cube  of  —  Ga-^J"  Jy^  ? 

Ans,    —  729a3*i3'(fy«, 

17.  What  is  the  fourth  power  of  —  4a^Pc*d^  ? 

Ans,   256a205i2ci6^2o 

18.  What  is  the  cube  of  —  Aa'^'^b^^'c^d? 

A71S.    —  64a«'"^»6«g9^j 

19.  What  is  the  fifth  power  of  2a^b^xy  ? 

Ans,   ^2a^^h^^^y^. 

20.  What  is  the  square  of  20a?»y"c*?   Ans,   400a;2'»?/2'«c^c^ 

21.  What  is  the  fourth  power  of  Sa^^^n^s? 

Ans.  81a*"68»c'2. 

22.  What  is  the  fifth  power  of  —  c'^d^^x^y'^  ? 

Ans,    —  c*'«(?i5«^b102/^^  v»u/vifw  - 

23.  What  is  the  sixth  power  of  —  a'^b'^^c*  ?  ^^    Z^ 

24.  What  is  the  fourth  power  of  —  laH'^d^, 

Ans,   16a«c«(?". 

POWERS      OF      FRACTIONS. 


lao.    From  the  definition  of  a  power,  and  the  rule  for 
e  mult 
written, 


the  multiplication  effractions,  the  cube  of  the  fraction  ^,  Is 


(aV  _  a      a      a  _  a^ 
b)  -  b  ^  I  ^  b  ^  b^' 


lao.  What  is  the  rule  for  raising  a  fraction  to  any  power  t 


164  ELEMENTARY       ALGEBRA. 

and  since  any  fraction  raised  to  any  power,  may  be  written 
under  the  same  form,  we  find  any  power  of  a  fraction  by 
the  following 

RULE. 

liaise  the  numerator  to  the  required  poicer  for  a  new 
numerator,  and  the  denominator  to  the  required  power  f<yr 
a  new  denominator. 

The  rule  for  signs  is  the  same  as  in  the  last  article. 

EXAMPLES 

Find  the  powers  of  tbe  following  fractions : 

/a— c\2  a^  —  2ac  +  c^ 

2      {  :^^^\  ,  Ant      ^1-, 


3.     (-r-V^h  A71S.       "^'^^ 


IQa'^b^ 


\Sbcl 

\    2ab    I' 

8.  Fourth  power  of  --r-;;                          -4w«.    , — r—  < 

^  2a;2y2                                          16j;V 

9.  Cube  of    — — ^ .       Jlws.   — ~-^- ^- "~ . 


P  O  W  JC  U  8      OF      B  I  N  O  M  I  A  L  rt .  1C5 


10.    Fourth  power  of •  -4;?^'.    ^^   .„  .^ 


11.    Fifth  power  of   —  ri; — -•  Ans.    —  ■zTr-rr'^' 

POWEES      OF      BINOMIALS. 

1 

121.    A  Binomial,  like  a  monomial,  may  be  raised  to  any 
power  by  the  process  of  continued  multiplication. 

1.  Find  the  fifth  power  of  the  binomial    a  i-  b. 
a  +    b Ist  power. 


a  +    b 

a^-^    ab' 

-h    ab  +  b 

2 

a?-  +  2ai  -h  &2 

.     .     2d 

a  +    5 

a3  +  la^b  -f 

ab^ 

-h    a^b  + 

2ab^   4- 

b^ 

a^  -f-  Zd}b  + 

3a62   4- 

b^    .    . 

.     .     3d 

a  -^    b 

a*  +  3a^b  + 

3a262  4- 

ab' 

+    a^b  4- 

3a2i2  4- 

Sab^   4- 

b* 

a*  -T   4a36  H- 

Qa'b^  4- 

4a*3  4- 

b*      4th 

a  +    5 

a*  4-  4a*b  4- 

6a362  4- 

4a2ft3  4. 

ab"^ 

4-    a*6  4- 

ia-'i^  4- 

6a2J3  ^  40^1  ^  j5 

a»  4-  6a»J  4-  lOa^ft*  4-  lOa^^  4-  6a6*  4-  ^     ^^w. 


H\  How  may  a  binomial  be  raised  to  any  power? 

122.  How  doe6  the  number  of  multiplications  compare  with  the  eX' 
poncnt  of  the  power?  If  the  exponent  is  4,  what  is  the  namber  of 
mnltiplications?  How  many  when  it  is  m?  How  many  things  arc  coo* 
sidered  in  the  raising  of  powers  ?    Name  them. 


166  ELEMENTARY       ALGEBliA.  ^ 

Note. — 122.  It  will  be  observed  that  the  number  of 
multiplications  is  always  1  less  than  the  units  in  the  expo* 
nent  of  the  power.  Thus,  if  the  exponent  is  1,  no  multipli' 
cation  is  necessary.  If  it  is  2,  we  multiply  once  ;  if  it  is  3, 
twice ;  if  4,  three  times,  &c.  The  powers  of  pol}Tiomials 
may  be  expressed  by  means  of  an  exponent.  Thus,  to 
ex:press  that  a  +  b  is  to  be  raised  to  the  5th  power,  we 
write 

{a  +  bY; 

if  to  the  mth  power,  we  write 

{a  +  by. 

2.  Find  the  5th  power  of  the  binomial  a  —  b. 

a  —    b     ,    ,    ^ 1st  power. 

a  —    b 


ab 

ab   +  b'' 


a^  —  2ah   4-  ^^ 2d  power. 

a  —      b 


t3  —  Q.a^b  4-    ah"^ 
—    a^b  4-     2a52 


«3  __  3^25  _|_    3^52   _  j3     ....     3d  power. 

a  —  b 

«4  _  za?h  +    ^aW  —      ab^ 
—    a'b  4-    3^252  _    3a53  4.  54 

(j4  _  4^35  ^    6a2Z>2  _    Aa¥  4-  **    •    4th  power. 
a  -b 


at  -  Aa'b  4-    Qa^b^  -    4^253  ^    ab* 

._    a*b  +    4aW  -    Qa'b^  4-  ^ab*  -  b" 
(jfi  -.  5a*b  +  lOaW  —  lOa^b^  4-  5«i*  —  b^'       Am, 


P0WRB6      OF      BINOMIALS.  167 

In  the  same  way  the  higher  powers  may  be  obtained.  By 
examining  the  powers  of  these  binomials,  it  is  plain  that  four 
thmgs  must  be  considered : 

Ist.  The  number  of  terms  of  the  power. 
2d.    The  signs  of  the  terms. 
3d.   The  exponents  of  the  letters. 
4th.  The  coefficients  of  the  terms. 

Let  us  see  according  to  what  laws  these  are  formed. 

Of  the  Terms, 

123.  By  examining  the  several  multiplications,  we  shall 
observ^e  that  the  first  power  of  a  binomial  contains  two  terms; 
the  second  power,  three  terms ;  the  third  power,  four  terms ; 
the  fourth  power,  five ;  the  fifth  power,  six,  &c. ;  and  hence 
we  may  conclude : 

TJiat  the  number  of  terms  in  any  power  qf  a  hinomial^ 
is  greater  by  one  than  the  exponent  of  the  power. 

Of  the  Signs  of  the  Terms, 

12<l.  It  is  evident  that  when  both  terms  of  the  ^ven 
binomial  are  plus,  aU  the  term^  of  the  power  will  be  plus. 

If  the  second  term  of  the  binomial  is  negative,  then  all 
the  odd  terms,  counted  from  the  left,  will  be  positive,  and 
aU  the  even  terms  negative. 


128.  How  many  terms  docs  the  first  power  of  a  binomial  contain  ?  The 
second  ?     The  third  ?    The  nth  power  ? 

]24.  If  both  terms  of  a  binomial  are  positive,  what  will  be  the  signs 
of  the  terms  of  the  power  ?  If  the  second  term  is  nogative,  how  are  the 
signs  of  the  terms  ? 


168  ELEMENTARY       ALGEBKA. 


Of  the  Exponents. 

125.  Tlie  letter  which  occupies  the  first  place  in  a  bino- 
mial, is  called  the  leading  letter.  Thus,  a  is  the  leading 
letter  in  the  binomials  a  -\-  h^  and  a  —  h. 

1st.  It  is  evident  that  the  exponent  of  the  leading  letter 
in  the  first  term,  will  be  the  same  as  the  exponent  of  the 
power ;  and  that  this  exponent  will  diminish  by  one  in  each 
term  to  the  right,  until  we  reach  the  last  term,  when  it  will 
be  0  (Art.  49). 

2d.  The  exponent  of  the  second  letter  is  0  in  the  first 
term,  and  increases  by  one  in  each  term  to  the  right,  to  the 
last  term,  when  the  exponent  is  the  same  as  that  of  the  given 
power. 

3d.  The  sum  of  the  exponents  of  the  two  letters,  in  any 
term,  is  equal  to  the  exponent  of  the  given  power.  Tliis 
last  remark  will  enable  us  to  verify  any  result  obtained  by 
means  of  the  binomial  formula. 

Let  us  now  apply  these  pruiciples  in  the  two  followmg 
examples,  m  which  the  coefficients  are  omitted : 

{a  +  hf  .   .  .  ««  +  a*5  +  a'lP'  +  aW  +  a^^*  _j_  ah^  +  ¥, 
\a  —  hY  .   .   .  a6  -  a^h  +  a'^^  —  a^h"^  +  a^^*  -  ab'>  +  &«. 

As  the  pupil  should  be  practised  in  writing  the  terms  with 
their  proper  signs,  without  the  coefficients,  we  will  add  a 
few  more  examples. 

125.  "Which  is  the  leading  letter  of  a  binomial?  What  is  the  exponent 
of  this  letter  in  the  first  term  ?  "  How  does  it  change  in  the  terms  towards 
the  right  ?  What  is  the  exponent  of  the  second  letter  in  the  second  term  ? 
How  does  it  change  in  the  terms  towards  the  right  ?  What  is  it  in  the 
last  term  ?    What  is  the  sum  of  the  exoonents  in  any  term  equal  to  ? 


POWERS      OF      BINOMIALS.  169 

1.  {a-\-by  .  ,a'-\-a^+ab^  f    b\ 

2.  (a  -by  .  ,  a*-a^b-^a^b^—  ab^  +  b\ 

8.  ia  +  bY_,  .  a''-^a'b-\ra'b'^-{-a^b^^-ab'-  -\-    b\ 

4.  ((r-6)'.  ,a?-n<b-\-a''b^-a*b'-{-a'b'-a'^b^'\-ah^-'h''. 

Of  the  Coefficients, 

l»i6.  The  coefficient  of  the  first  terra  is  1.  The  coeffi 
cient  of  the  second  term  is  the  same  as  the  exponent  of  the 
given  power.  The  coefficient  of  the  third  term  is  found  by 
multiplying  the  coefficient  of  the  second  term  by  the  expo- 
nent of  the  leading  letter  in^that  term,  and  di\'iding  tho 
product  by  2.    And  finally : 

Jf  the  coefficient  of  any  term  t>e  multiplied  by  th^  expo- 
nent of  the  leading  letter  in  that  term,  and  the  product 
divided  by  the  number  which  marks  the  place  of  the  term 
from  the  left.  Vie  quotient  will  be  the  coefficie^it  of  the 
next  term. 

Thus,  to  find  th*»  coefficients  in  the  example, 

(a-by  ,  .  ,  a'-  a^b  4-  a^l^-a'b'-\-  d'b*-  a'^b''  +  ab^-  b\ 

we  first  place  the  exponent  1  as  a  coefficient  of  the  second 
term.  Then,  to  find  the  coefficient  of  the  third  term,  we 
multiply  7  by  6,  the  exponent  of  a,  and  divide  by  2.  The 
quotient,  21,  is  the  coefficient  of  the  third  tenn.  To  find  the 
coefficient  of  the  fourth,  we  multiply  21  by  5,  and  divide 
the  product  by  3  ;  this  gives  35.  To  find  the  coefficient  of 
tlic  fifth  term,  we  multiply  35  by  4,  and  divide  the  product 
by  4 ;  this  gives  36.    The  coefficient  of  the  sixth  tenn,  found 

126.  Wliat  is  the  coefficient  of  the  first  term  ?    What  is  the  coofficicnt 
ol  the  second  term  ?     IIow  do  you  find  the  coefficient  of  the  third  term 
How  do  you  find  the  coefficient  of  any  term  ?    What  are  the  coofficieiite 
of  the  first  and  last  terms  ?     How  are  the  coefficients  of  the  exponente 
ot  any  two  terms  equally  diftant  from  the  two  extremes T 
8 


170  KLEMENTAEY      ALGEBEA. 

in  the  same  way,  is  21 ;  that  of  the  seventh,  7 ;  and  that  of 
the  eighth,  1.     Collecting  these  coefficients, 

(a  -  hy  = 

a'  -  la^b  -I-  21a^b'^-35a^b^  +  ^5a^b*  -  21a'^b^  4-  7a6«  —  b\ 

Note.— We  see,  in  examining  this  last  result,  that  the 
voeffisients  of  the  extreme  terms  are  each  1,  and  that  the 
coefficients  of  terms  equally  distant  from,  the  extreme  terms 
are  equal.  It  will,  therefore,  be  sufficient  to  find  the  coeffi- 
cients of  the  first  half  of  the  terms,  and  from  these  the 
others  may  be  immediately  written. 

EXAMPLES 

1.  Find  the  fourth  power  of  a  +  b, 

Ans.  a*  +  Aa^b  +  ea^J^  +  ^a¥  +  &♦. 

2.  Find  the  fourth  power  oi  a  —  b, 

Ans.   a*  —  4a35  +  ^aW  —  4a63  +  5*. 

3.  Fmd  the  fiilh  power  of  a  +  5. 

Ans.   a^  +  bO'b  +  lOa^^^  ^  loa^i^  +  5a6*  +  ¥. 

4.  Find  the  fifth  power  of  a  —  J. 

Ans.   a^  —  ba^b  +  lOaW  —  lOa^ft'  ~\-  fiab'  —  ^. 

6.  Find  the  sixth  power  of  a  +  5. 

^6  +  6^55  4.  15^452  j^20a^b^  +  16«^6'  +  606^  +  66. 

8.  Find  the  sixth  power  of  a  —  b. 

127.  When  the  terms  of  the  binomial  haye  coefficients, 
we  may  still  write  out  any  power  of  it  by  means  of  the 
Binomial  Formula. 

7.  Let  it  be  requu-ed  to  find  the  cube  of  2c  +  Sd» 

(a  +  5)3  =  a3  +  ^^b  +  dab^  +  b^ 


POWBEfl      OF       BINOMIALS.  171 

Here,  2c  takes  the  place  of  a  in  the  formula,  and  Zd  the 
place  of  b.    Hence,  we  have, 

(2c+Zdy=:  (2c)3+3.(2c)2.3(;+3(2c)(3df)2+(3cf)3     .     (1; 

and  l>y  perfoiining  the  indicated  operations,  we  have, 

(2c  +  3J)3  =  8c3  +  3Cc^<Z  +  5icd^  +  2ld\ 

If  we  examine  the  second  member  of  Equation  ( 1 ),  we 
see  that  each  term  is  made  up  of  three  fiictors:  Ist,  the 
nimierical  factor;  2d,  some  power  of  2c;  and  3d,  some 
power  of  Zd,  The  powers  of  2c  are  arranged  in  descend- 
ing order  towards  the  right,  the  last  term  involving  the  0 
power  of  2c  or  1 ;  the  powers  of  3c?  are  arranged  in  ascend- 
ing  order  from  the  first  term,  where  the  0  power  enters,  to 
the  last  term. 

The  operation  of  raising  a  binomial  involving  coefficients, 
is  most  readily  effected  by  writing  the  three  factors  of  each 
term  in  a  vertical  column,  and  then  performing  the  multipli* 
cations  as  indicated  below.  ^ 

Find,  by  this  method,  the  cube  of  2c  +  3(f. 


OPERATION. 

1 

4- 

3 

+ 

3 

+ 

1 

Coefficients. 

8c3  + 

4c2 

+ 

2c 

+ 

1 

Powers  of  2c 

1 

+ 

Zd 

+ 

9^2 

4-  27J3 

Powers  of  3  c? 

(2c  -I-  dy  =  8c3  +  ^QcH  H-  54c<?2  +  2ld^ 

The  preceding  operation  hardly  requires  explanation.  la 
the  first  line,  \^Tite  the  numerical  coefficients  corresponding 
to  the  particular  power ;  in  the  second  line,  write  the  do- 
Bccnding  powers  of  the  leading  term  to  the  0  power ;  in  the 
third  line,  write  the  ascending  powers  of  the  following  term 
from  the  0  power  upwards.     Tl  will  be  easiest  to  conimonce 


172  ELEMENTARY       ALGEBRA. 

the  second  line  on  the  right  hand.    The  mnltipUcation  should 
be  performed  from  above,  downwards. 

8.  Fmd  the  4th  power  of  Sa\"  —  2bd. 

(a  +  by  =«*-!-  ^a^b  +  Qa^b^  +  4:ab^  +  b\ 

l-f-4  +6  +4  +1 

81  aV  +    27a«c2      +       9a*c2         +    Sa'^c         +    1 
1         _      2bd        +      4^2(^2        -    SbhP        +  16b'd  . 

81aV—  21Qa^c^bd+  2lQa'c^b''d^  —  dda'^cb^d^  +  lQb*d\* 

9.  What  is  the  cube  of  3aj  —  6y  ? 

Ans.    27ic3  —  162a;22/  +  324icy2  _  21Qy\ 

10.  What  is  the  fourth  power  of  a  —  3^? 

Ans.   a*  —  12a^b  +  54a2^,2  _  losab^  +  815*. 

11.  What  is  the  fifth  power  of  c  —  2d? 

Ans.   c^  —  lOc^d  +  40c^d^  —  SOc'^d^  +  80cc7*  —  S2d^. 

12.  What  is  the  cube  of  5a  —  3c?? 

Ans.    125^3  -  22oa^d  +  135«c?2  _  27(?^ 


•  This  ingenious  method  of  writing  the  development  of  a  binomial  ia   due  to 
Profcseor  William  G.  Fbok,  of  Columbia  CJollege. 


BXTHAOTION      OF       80  0  1S.  173 


CHAPTER    VIL 

SQUACB      BOOT.         EADICALS      OF     THK     SECOND 
DE6BEE. 

12§.  The  Square  Root  of  a  number  is  one  of  its  two 
equal  factors.  Tlius,  6x6  =  36;  therefore,  6  is  the  square 
root  of  36. 

The  symbol  for  the  square  root,  is  -y/  ,  or  the  fractional 
exponent  ^ ;  thus, 

v/a,    or    a  , 

indicates  the  square  root  of  a,  or  that  one  of  the  two  equal 
(actors  of  a  is  to  be  found.  The  operation  of  finding  such 
factor  is  called,  Extracting  the  Square  Root. 

129.  Any  number  which  can  be  resolved  hito  two  eqnal 
integral  factors,  is  called  a  perfect  square. 

The  following  Table,  verified  by  actual  multiplication,  in- 
dicates all  the  perfect  squares  between  1  and  100. 

TABLE. 

1,     4,     9,     16,     25,     36,     40,     64,     81,     100,     squares. 
1,     2,     3,      4,       5,       6,       7,       8,       9,       10,      roots. 

128.  What  ifl  the  square  root  cf  a  number  ?  Wlia  is  the  opcrsitioo  of 
finding  th^  equal  fiictor  (uilled  ? 

129.  What  is  a  perfect  square  ?  How  manj  perfect  squares  are  then? 
between  I  and  1»K),  incl  'ding  botF  numbers  T     What  are  they? 


174  ELEMENTARY      ALGEBRA. 

We  may  employ  this  table  for  finding  the  square  root  of 
any  perfect  square  between  1  and  100. 

Looh  for  the  number  in  tJie  first  line ;  if  it  is  found 
there^  its  square  root  will  he  found  immediately  under  it. 

K  the  given  number  is  less  than  100,  and  not  a  perfect 
square,  it  will  fall  between  two  numbers  of  the  upper  line,  and 
its  square  root  will  be  found  between  the  two  numbers  directly 
hdow  ;  the  lesser  of  the  two  will  be  the  entire  part  of  the 
root,  and  will  be  the  true  root  to  within  less  than  1. 

Thus,  if  the  given  number  is  55,  it  is  found  between  the 
perfect  squares  49  and  64,  and  its  root  is  7  and  a  decimal 
fraction. 

Note.— There  are  ten  perfect  squares  between  1  and  100, 
if  we  include  both  numbers ;  and  eight,  if  we  exclude  both. 

K  a  number  is  greater  than  100,  its  square  root  will  be 
greater  than  10,  that  is,  it  will  contain  tens  and  units.  Let 
.N'  denote  such  a  number,  x  the  tens  of  its  square  root,  and 
y  the  units ;  then  will, 

JSr  =  (x  +  yY  =  x^  -^  2xy  -h  y^  =  x^  +  {2x  +  2/)y. 

That  is,  the  number  is  equal  to  the  square  of  the  tens  in  its 
roots,  plus  twice  the  product  of  the  tens  by  the  units,  plus 
the  square  of  the  units. 

EXAMPLE. 

1.  Extract  the  square  root  of  6084. 

Since  this  number  is  composed  of  more  than 
two  places  of  figures,  its  root  will  contain  more  60  84 

than  one.    But  since  it  is  less  than  10000,  which 
is  the  square  of  100,  the  root  will  contain  but  two  figures' 
that  is,  unit8  and  tens. 


SQUARE   ROOT   OF   NUMBERS.       175 

Now,  the  square  of  the  tens  must  be  found  in  the  two 
left-hand  figures,  which  we  will  separate  from  the  other  two 
by  putting  a  point  over  the  place  of  units,  and  a  second  over 
the  place  of  hundreds.  These  parts,  of  two  figures  each, 
are  called  periods.  The  part  60  is  comprised  between  the 
two  squares  49  and  64,  of  which  the  roots  are  V  and  8 ;  hence, 
7  expreeses  the  number  of  tens  sought ;  and  the  required 
root  is  composed  of  7  tens  and  a  certain  number  of  units. 

The  figure  1  being  found,  we 
write  it  on  the  right  of  the  given  60  84 

number,  from  which  we  separate  49 


78 


it  by  a  vertical  line  :  then  we  Y  X  2  =  14  8  ]  118  4 

subtract  its  square,  49,  from  60,  118  4 

which  leaves  a  remainder  of  11,  0 
to  which  we  brinsr  down  the  two 


*o 


next  figures,  84.  The  result  of  this  operation,  1184,  con- 
tains twice  the  product  of  the  tens  by  the  units,  plus  the 
square  of  the  units. 

But  since  tens  multiplied  by  units  cannot  give  a  product 
of  a  less  unit  than  tens,  it  follows  that  the  last  figure,  4,  can 
form  no  part  of  the  double  product  of  the  tens  by  the  units , 
this  double  product  is  therefore  found  in  the  part  118,  which 
we  separate  from  the  units'  place,  4. 

Now  if  we  double  the  tens,  which  gives  14,  and  then 
divide  118  by  14,  the  quotient  8  will  express  t/ie  units,  or  a 
number  greater  than  the  units.  This  quotient  can  never  be 
too  small,  smce  the  part  118  will  be  at  least  equal  to  twice 
the  product  of  the  tens  by  the  imits ;  but  it  may  be  too 
large,  for  the  118,  besides  the  double  product  of  the  tens  by 
the  units,  may  likewise  contain  tens  arising  from  the  square 
of  the  units.  To  ascertain  if  the  quotient  8  expresses  the 
right  number  of  units,  we  write  the  8  on  the  right  of  the  14, 
which  gives  148,  and  then  we  multiply  148  by  8.  This 
multiplication  being  effected,  gives  for  a  product,  1184.  a 


176  ELEMENTARY       ALGP:BKA. 

number  equal  to  the  result  of  the  Urst  operation.  Hav- 
ing subtracted  the  product,  we  find  the  remainder  equal 
to  0  ;  hence,  78  is  the  root  required.  In  this  operation, 
we  form,  1st,  the  square  of  the  tens;  2nd,  the  double 
product  ot  the  tens  by  the  units;  and  3d,  the  square  of 
the  units. 

Indeed,  in  the  operations,  we  have  merely  subtracted  froip 
the  given  number  6084  :  1st,  the  square  of  V  tens,  or  of  70; 
2d,  twice  the  product  of  70  by  8 ;  and,  3d,  the  square  of  8 ; 
that  is,  the  thj-ee  parts  which  enter  into  the  composition  of 
the  square,  70  +  8,  or  78  and  since  the  result  of  the  sub- 
traction is  0,  it  follows  tha'.  78  is  the  square  root  of  6084. 

ISO.  The  operations  in  the  last  example  have  been  per- 
formed on  but  two  periods,  but  it  is  plain  that  tlie  same 
nif'thods  of  reasoning  are  equally  applicable  to  larger  num- 
bers, for  by  changing  the  order  of  the  units,  we  do  not 
change  the  relation  in  which  they  stand  to  each  other. 

Thus,  in  the  number  60  84  95,  the  two  periods  60  84, 
havo  the  same  relation  to  each  other  as  m  the  number 
60  84  ;  and  hence  the  methods  used  in  the  last  example  are 
equally  applicable  to  larger  numbers. 

131.  Hence,  for  the  ex-traction  of  the  square  root  of 
numbers,  we  have  the  followmg 

RULE. 

I.  Point  off  the  given  number  into  periods  of  two  figures 
each^  beginning  at  the  right  hand: 

II.  Note  the  greatest  perfect  square  in  the  first  period  on 
tlie  kfty  and  place  its  root  on  the  right,  after  the  manner  of 

ISl.  Give  the  rule  for  the  extracti(»n  of  the  square  roDt  of  nnrabers? 
What  18  the  first  step  ?  What  the  sec'iiid  ?  What  the  third  ?  What  the 
fourth?     What  the  filth? 


SQUARE      ROOT      OF      NUMBERS.  177 

a  qiiotiefit  in  division  ;  then  subtract  tlie  square  of  thin 
root  from  the  first  period^  and  bring  down  the  second  period 
for  a  remainder  : 

in.  Double  tJie  root  already  founds  and  place  tlie  result 
on  the  left  for  a  divisor.  Seek  how  many  times  the  divisor 
is  catJaified  in  tJte  remainder^  exclusive  of  the  right-hand 
fiffure^  and  plfice  the  figure  in  the  root  and  also  at  the  righi 
of  the  divisot  : 

rV.  Multiply  the  divisor^  thus  atigmented^  by  the  last 
figure  of  the  root,  and  subtract  tJie  product  from  the  re- 
mainder^ and  bring  dozen  tJie  next  period  for  a  new  remaifi- 
der.  But  if  any  of  the  products  should  be  greater  tlian 
the  remainder^  diminish  the  last  figure  of  the  root  by  one  : 

V.  Dozible  the  whole  root  already  fouyid^  for  a  new  di- 
visor,  and  co?itinue  the  operation  as  before,  until  all  tJie 
periods  are  brought  down. 

132.  Note. — 1.  If,  after  all  the  periods  are  brought 
down,  there  is  no  reniauider,  the  given  number  is  a  perfect 
square. 

2.  Tlie  number  of  places  of  figures  in  the  root  will  always 
be  equal  to  the  number  of  periods  into  which  the  given 
number  is  divided. 

3.  If  the  given  number  has  not  an  exact  root,  there  will 
be  a  remainder  after  all  the  periods  are  brought  do\\Ti,  in 
which  case  ciphers  may  be  annexed,  forming  new  periods, 
for  each  of  which  there  will  be  one  decimal  place  in  the  root. 


132.  What  Ukes  place  when  the  given  number  is  a  perfect  square  f 
How  many  places  of  figures  will  there  be  in  the  root?  If  the  given  num. 
bor  \b  not  a  perfect  square,  w  hal  may  Se  done  after  ill  the  periods  are 
brought  down  7 


178 


ELEMENTARY      ALGEBRA 


EXAMPLES. 

1.  What  is  the  square  root  of  36729  ? 


In  this  example  there  are 
two  periods  of  decimals, 
and,  hence,  two  places  of 
decimals  in  the  root. 


3  67  29,191.64+ 
1 


29 


267 
261 


38  1 

629 
381 

382  6 

24800 
22956 

3832  4 

184400 
153296 

31104  Rem. 


2.  To  find  the  square  root  of  7225. 

3.  To  find  the  square  root  of  17689. 

4.  To  find  the  square  root  of  994009. 

5.  To  find  the  square  root  of  85673536. 

6.  To  find  the  square  root  of  67798756. 

7.  To  find  the  square  root  of  978121. 

8.  To  find  the  square  root  of  956484. 

9.  What  is  the  square  root  of  36372961  ? 

10.  What  is  the  square  root  of  22071204? 

11.  What  is  the  square  root  of  106929? 

12.  What  of  12088868379025  ? 

13.  What  of  2268741  ? 

14.  What  of  7596796? 

1 5.  What  is  the  square  root  of  96  ? 

16.  What  is  the  square  root  of  153  ? 

17.  What  is  the  square  root  of  101  . 


Ans.  86. 
Ans.  133.. 
Ans.  997.  / 
Ans.  9250.    ' 
Ans.  8234. 
Ans.  989. 
Ans.  978. 
Ans.  6031. 
Ans.  4698. 
Ans.  327. 
Ans.  3476905. 
Ans.  1506.23  + 
Ans.  2756.22  -f 
A71S.  9.79795    f- 
Ans.  12.36931  4  . 
Ans.  10.0 }  087    f , 


6QUAEE     EOOT     OF     FBAOTIONB.  179 

■  18.  What  of  285970396644?  Am,  634762. 

19.  What  of  41605800625  ?  Ans.  203975. 

20.  Wliat  of  48303584206084?  Ans.  6050078. 

BSITBACnON   OP  THE  SQUAEB  EOOT   OP  PBACnONS. 

133.  Since  the  square  or  second  power  of  a  fraction  iw 
obtained  by  squaring  tlie  numerator  and  denominator  sepa- 
rately, it  follows  that 

The  sqtiare  root  of  a  fraction  wiU  he  equal  to  the  square 
root  of  tJie  numerator  divided  by  the  sqicare  root  of  the 
denominator. 

For  example,  the  square  root  of  ^    is  equal  to   ^ :  for, 
a      a  _a^ 

I  1 

1.  What  is  the  square  root  of  -?  Ans,  -• 

9  3 

2.  What  is  the  square  root  of   — -  ?  Ans.   -  • 

16  "4 

8.  What  is  the  square  root  of  —  ?  Ans.   -  • 

81  9 

256  16 

4.  What  is  the  square  root  of   — —  ?  Ans,  —  • 

1  ft  1 

5.  What*  is  the  square  root  of  —  ?  Ans,  -• 

At\Ci(K  t^A. 

6.  What  is  the  square  root  of  ?  Ans,  -— . 

olOUy  ^47 

H    ^^    .'    .X.  *    ^   582169^  .         763 

7.  \Viiat  IS  the  square  root  of  ?  Ans,  -—  • 


1A8.  Td  what  is  the  square  root  of  a  fracticm  equal  f 


180  ELKMENfAEY        ALGEBKA. 

1^4.  II*  the  numerator  and  denominator  are  not  perfect 
squares,  the"  root  of  the  fraction  cannot  be  exactly  found. 
We  can,  however,  easily  find  the  approximate  root. 

RULE. 

Multiply  both  terms  of  the  fraction  by  the  denominator  : 
Then  extract  the  square  root  of  the  numerator^  and  divide, 
this  root  by  the  root  of  the  denominator  ^  the  quotient  wiM 
be  the  approximate  root, 

1.  Find  the  square  root  of  -  • 

o 

Multiplying  the  numerator  and  denominator  by  6 


hence,     (3.8729  +)  ->  5   =     .7745  +   =  Ans, 

7 

2.  What  is  the  square  root  of   -  ?  Ans.    1.32287  +. 

•  * 

14 

3.  W^hat  is  the  square  root  of  —  ?         Ans.   1.24721  +, 

y 

4.  What  is  the  square  root  of   Utt?     ^^5«   3.41869  -}-. 

1  Q 

5.  What  is  the  square  root  of   7— 7  ?       A71S.   2.71313  +. 

36 

6.  What  is  the  square  root  of   8—  ?       Ans.   2.88203  +.    ^ 

7.  What  is  the  square  roct  of   ~  ?         -^ns.   0.64549  -f .   d 

Q 

8.  What  is  the  square  root  of    lO—  ?     Ans.   3.20936  -f . 

134.  What  is  the  rule  when   the  numerator  atd  denominator  arc  not 
5>erfect  squares  ? 


6QDAKE      ROOT     OF     MONOMIALS.  181 

135.  Finally,  insteafl  of  the  last  method,  we  may,  if  wo 
please, 

Change  the  common  fraction  into  a  decimal,  and  continue 
the  division  until  the  7iumher  of  decimal  places  is  double 
the  number  of  places  required  in  tJie  root  Then  extract 
the  root  of  the  decimal  by  the  last  rule, 

BZAMPLES. 

1.  Extract  the  square  of   —    to  within  .001.     Thisnum 

14 

ber,  reduced  to  decimals,  is  0.786714  to  within  0.000001 ;  but 
the  root  of  0.785714   to  the  nearest  unit,  is   .886;  hence, 

0.P86  is  the  root  of   —    to  witliin  .001. 

2.  Find  the    \/2—   to  within  0.0001.     A7is.   1.6931  +. 

V     15 

3.  What  is  the  square  root  of  r^  ^         ^^'   0.24253  +. 

7 

4.  What  is  the  square  root  of   -  ?  Ans,   0.93541  +. 

8 

5 

6.  What  is  the  square  root  of   -  ?  Ans.   1.29099  -\-, 


EXTRACTION   OP  THE  SQUARE  ROOT   OP   MONOMIALS. 

136.  In  order  to  discover  the  process  for  extracting  the 
Bquarc  root  of  a  monomial,  we  must  see  how  its  squaie  \k 
formed. 

By  the  rule  for  the  multiplication  of  monomials  (Art.  42)t 
we  liave, 

{ba'^b'^cY  =  ^o^h^c  X  oa^b'^c  =  25a*Z»«c2 ; 

186.  What  is  a  second  method  of  finding  the  approximate  root? 
186.  Give  the  rule  for  extracting  the  squar**  root  of  mouomials? 


182  ELEMENTARY      ALGEBRA. 

that  is,  in  order  to  square  a  monomial,  it  is  necessary  to 
square  its  coefficient  and  double  the  exponent  of  each  of  the 
letters.  Hence,  to  find  the  square  root  of  a  monomial,  we 
have  the  following 

RULE. 

1  Bxtract  tJie  square  root  of  the  coefficient  for  a  new 
coefficient : 

n.  Divide  the  exponent  of  each  letter  by  2,  and  then 
annex  all  the  letters  with  their  new  exponents. 

Since  like  signs  in  two  factors  give  a  plus  sign  in  the  pro- 
duct, the  square  of  —  a,  as  well  as  that  of  +  «,  will  be 
+  a^\  hence,  the  square  root  of  a^  is  either  +  a,  oi 
—  a.  Also,  the  square  root  of  25a'^64,  is  either  +  hab'^y 
or  —  hab"^.  Whence  we  conclude,  that  if  a  monomial  is 
positive,  its  square  root  may  be  affected  either  with  the  sign 
4-  or  —  ;  thus,  -/Oo*  =  ±3^2;  for,  +  Sa^  or  —  Sa^, 
squared,  gives  -\-  9a*.  The  double  sign  ±,  with  which  the 
root  is  affected,  is  read  plus  and  minus, 

EXAMPLES. 

1.  What  IS  the  square  root  of   64a^6*? 

V'64a65*  z:^  ^-%aW\  for  -^^aWx  +8a3J2_  _|_64a65* 
and,  Veia^^:  -%am\  for  -^a^b'^y.  -Sa^b^z=  +64a«6* 
Hence,  y^Ia^  =   ±  8a^bh 

2.  Find  the  square  root  of  Q25aWc^.  ±  25a¥c\ 

3.  Find  the  square  root  of  blQa^¥c^.  ±  ^ia^^cK 

4.  Find  the  square  root  of  196a;y2*.  ±  l^xhjz^. 

5.  Find  the  square  root  of  Ula^b^c^^d^^  ±  ^la^'b^c^d^ 

6.  Find  the  square  root  of  lS4a^W^c^^d^  ±  2Sa^b''c^d, 
Y.  Find  the  square  root  of  81«^i*c^  ±  9a'* />V. 


IMPERFECT      SQUARK6.  188 

Notes. — 137.  1.  From  tho  preceding  rule  it  follows, 
that  wbeu  a  monomial  is  a  perfect  square,  its  numerical 
coefficient  is  a  perfect  square^  and  all  its  exponmts  even 
numbers.    Thus,    25a*b^    is  a  perfect  square. 

2.  If  the  proposed  monomial  were  negativCy  it  would  be 
impossible  to  extract  its  square  root,  since  it  iias  just  been 
shown  (Art.  136)  that  the  square  of  every  quantity,  whether 
positive  or  negative,  is  essentially  positive.    Therefore, 

are  algebraic  symbols  which  indicate  operntions  that  cannot 
be  performed.  Tlicy  are^ called  imaginai  </  quantities^  or 
rather,  imaginary  expressions,  and  are  fiequently  met  with 
In  the  resolution  of  equations  of  the  second  degree. 


IMPERFECT  SQUARES. 

138.  Wlien  the  coefficient  is  not  a  perfect  square,  or 
vvheu  the  exponent  of  any  letter  is  uneven,  the  monomial  is 
an  imperfect  square :  thus,  98ab*  is  an  imperfect  square. 
Its  root  is  then  indicated  by  means  of  the  -adical  sign ;  thus. 


Such  quantities  are  called,  radical  quantities,  or  radicals  of 
the  second  degree :  hence, 

A  RADICAL  QUANTITY,  is  the  indicated  root  of  an  imperfect 
power. 


137.  When  ia  a  monomial  a  perfect  sqnare?  What  monomiala  are 
these  whose  square  roots  cannot  be  extracted  ?  What  are  such  ezpre*' 
tioos  called  ? 

188.  When  is  a  monomial  an  imperfect  square  ?  What  are  such  qr/ui 
tlliHS  CJillcil  ?     Wljat  ifl  A  rrulicrti  quantity  t 


184  elementary'  algebr 


TRANSFORMATION       OF      RADICALS. 

139.  Let   a   and   h   denote   any  two  numbers,  and  p 
the  product  of  their  square  roots:  then, 

-/«  X   V^   r=  jo (1.) 

Squaring  both  members,  we  have, 

a  X  b  ^  p^      ....     (2.) 

Then,  extracting  the  square  root  of  both  members  of  (2  ), 

■y/ab  —  p (8.) 

And  since  the  second  members  are  the  same  in  Equations 
( 1 )  and  (  3  ),  the  first  members  are  equal :  that  is. 

The  square  root  of  the  product  of  two  quantities  is  equal 
to  the  product  of  their  square  roots. 

140.  Let  a  and   b  denote   any  two  numbers,  and   q 
the  quotient  of  their  square  roots ;  then, 

1=' « 

Squaring  both  members,  we  have, 

1  =  2^ (2.) 

then  extracting  the  square  root  of  both  members  of  ( 2 ), 

f  =  ^ (^) 

and  since  the  second  members  are  the  same  in  Equations  ( 1 ) 
and  ( 3  ),  the  first  members  are  equal ;  that  is, 

139.  To  what  is  the  square  root  of  the  product  of  two  quantities  eqiial? 

140.  To  what  is  the  square  root  of  the  quotient  of  two  quactiticE 
equal  ? 


TKANSFOHMATION      OF      RADICALS.       135 

T?ie  square  root  of  the  quotient  of  two  quantities  is  equal 
to  the  quotient  of  their  square  roots. 

These  principles  enable  us  to  transform  radical  expres- 
sions, or  to  reduce  them  to  simpler  forms ;  thus,  the  expres- 
sion, 

98aJ*  =  495*  X  2a ; 

hence,  -/98a6*  =:   v'496*  x  2a; 

and  by  the  principle  of  (Art.  139), 

V495*  X  2a  =    -v/49^*  X   V^  =   Wy/^. 
In  like  manner, 
^/Abc^i^^  =   -/9a2^>2c2  X  bbd       =     Zahc^/bhd, 
-/864a2^»Vi  =   '/I44a"'=^»*ci"  X  Qbc  =  Uab'^c^ y/ebc. 

The  COEFFICIENT  of  a  radical  is  the  quantity  without  the 
sign ;  thus,  in  the  expressions, 

7^V2a,      ^abc^/5bdy      12a5VV^, 

the  quantities   7^^    3a5c,    12aJV,    are  coefficieJits  of  the 
radicals, 

141.  Hence,  to  simplify  a  radical  of  the  second  degree, 
we  have  the  following 

BULE. 

I.  Divide  the  expression  under  the  radical  sign  into  two 
factorSy  one  of  which  shall  be  a  perfect  square  : 

TL  Extract  the  square  root  of  the  perfect  square^  and 
then  multiply  this  root  by  the  indicated  square  root  of  ths 
remaining  factor, 

141,  Give  the  rule  for  simplifying  md^.^als  of  the  second  degree.  How 
do  you  deiermine  whether  a  given  n  :  .\  factor  which  is  a  perfect 

square? 


186 


ELEMENTARY       ALGEBKA 


Note. — ^To  determine  if  a  given  number  has  any  factor 
which  is  a  perfect  square,  we  examine  and  see  if  it  is  divi- 
sible by  either  of  the  perfect  squares, 

4,     9,     16,     25,     36,     49,     64,     81,  &c.; 

tf  it  is  not,  we  conclude  that  it  does  not  contain  a  fiictcr 
which  is  a  perfect  square. 

EXAMPLES. 

Reduce  the  following  radicals  to  their  sunplest  form : 


1.  ^5d^hc. 


2.  ^/l2^h^ahP. 

3.  ^S2aF¥c. 


Ans.  ba-^Zahc. 
Ans.  W-a^dJ2}). 


4.    ^/2h^d^l^C^, 


5.  ^/^m^(F8^, 

6.  \/l2^a'h^c^d. 


1,  ^i5a?b^cH, 
8.  y/lUba^cM^ 

10.  V^TsOoio^V. 

11.  ^Oba/b^d^. 


Ans.  4a*6*  \/'2aG, 

Ans^lQcfPc\ 

Ans.  Z2a'^Pc- -^/abc.' 

Ans.  2la'^h'^c^\^abd. 

Ans.  15aWc^abd. 

Ans.  l^cfc'^d^^/da. 

Ans.  12a*d^m'^^7<yd. 

Ans.  l^a^b^c^ \/ll, 

Ans.  9aWd\/^. 


142.  Notes. — 1.  A  coefficient^  or  a  factor  of  a  coeffi 
cient,  may  be  carried  under  the  radical  sign,  Jy  sqicariiig  it. 
Thus, 

1.  Za^^/bc  =   ^Sa'Y  x  be  =  ^a^e. 

2.  2ab^/d   =■  2y/aWd  =  ^a'^b^d. 


142.  How  may  a  coefficient  or  factor  be  carried  under  the  radical  Figii 
To  what  is  the  square  root  of  a  uegat'Ve  quantity  equal  ? 


ADDITION       OP      RADICALS 


187 


8.  4(a+b)y^a^=4y/{a+iy(a-b)=:W(a'^b^)(a-hb) 

2.  The  square  root  of  a  negative  quantity  may  also  be 
simplified;  thus, 

-/31  =  V9  X  -  1  =  y^  X  -/^^  —  3^v/-~i, 

and,       -/—  4a2  =   -y/io^  x  -/—  1  =  2aV—  1 ;       also 

^/—8a^b  =  \/4a^x  —2*  =  2aV-2^  =  2flr y^  X  -/^ ; 

that  is,  the  square  root  of  a  negative  quantity  is  equal  to 
the  square  root  of  the  same  quantity  with  a  positive  sign^ 
multiplied  into  the  square  root  of  —  \, 

Reduce  the  following : 


1.  ■/— 64a2^>2. 


2.  V—  128a*6^ 

3.  V—  na^h''&, 

4.  y/-  48a?b(f, 


Ans.  ^ah^  — ^. 


ADDITION    OF    RADICALS. 

143.  SnoLAE  Radicals,  of  the  second  degree,  are  those 
in  which  the  quantities  under  the  sign  are  the  same.  Tims, 
the  radicals  3y^,  and  bc^  are  similar,  and  so  also  are 
9v/2,  and  1'/2. 

144.  Radicals  are  added  like  other  algebraic  quantities 
hence,  the  following 

148.  What  are  similar  radicals  of  the  second  degree? 

144.  Give  th<  rule  for  the  addition  of  radicals  of  the  eccond  degree  ? 


188  ELEMENTARY      ALGEBRA 

KFLE. 

I.  If  the  radicals  are  similar^  add  their  coefficients^  and 
to  the  sum  annex  the  common  radical : 

n.  If  the  radicals  are  not  similar^  connect  them  together 
with  their  proper  signs. 

Thus,         3a v^  +  bc^/h  —  (3a  -h  5c)  v^. 
In  like  manner, 

7v'2a  +  Sy^  =  (7  +  3)V5a  =  10^/2a. 

Notes. — 1.  Two  radicals,  which  do  not  ajjpear  to  be  sim- 
ilar at  first  sight,  may  become  so  by  transformation  (Art. 
141.) 

For  example, 


V48aP  +  h^/Wa  =  4:b\/da  +  5b^3a  =  Qb^/Sa; 
2v^  +  3-/^    =   Q^/5  +  S^   =   9^5. 

2.  When  the  radicals  are  not  similar,  the  addition  or  sub- 
traction can  only  be  indicated.  Tims,  in  order  to  add  3  \/h 
to  5-v/«,  we  write, 

5ya  4-  3v^. 

Add  together  the  following : 

1.  'v/27a2   and    x/48a\  Ans.  *lay/^, 

2.  ^/hWd^   and    ^2af¥.  Ans,  Ua^b^/2. 

3.  v/— -    and  \/-r-'  Ans.  4a^ 
V     5                 V   15 

4.  ^125    and    y^OOa^.  Aiis,  (5.-{-  10a)  y^ 
^      /5o"       ,      /Too  .       10   r- 


8UBTBA0TI0N      OF      BADI0AL8.  189 


6.  V98a2«  and    ySCar^  —  3Qa\ 

Ans,  YayS  +  6Vaj2__  a\ 

1.  -v/OSa^  and    v^288a*x*.      -4n«.  {la -^  I2ah:'^)'/2x. 

J    8.  V^  and    -/TSs. 

9.   -^27   and    -/liV- 


-4n5. 

U^. 

-4?is. 

lOv/2; 

^    Ans, 

30^ 

3,  (2a  4-  24aj2)y^. 

Ans. 

119-v/3. 

11.  2V^   and  3-/64&C*. 

12.  v^43    and   loVsca. 

13.  y/^20a^^  and    ^2450^.      ^W5.  (8aJ  +  7a*53)v^ 

14.  v^5a*^   and   -/SoOo^.  -4n«.  {5a^b^ -i-  10^3^.2)^5; 


D  suBTRAcnorr  of  eadioai^. 


145,  Radicals  are  subtracted  like  other  algebraic  quan- 
tities ;  henre,  the  following 

EU  LE. 

I.  If  the  radicals  are  similar,  subtract  the  coefficient  of 
the  subtraJiend  from  that  of  the  minuend^  and  to  t/ie  differ^ 
eiice  annex  the  common  radical : 

n.  Jf  the  radicals  are  not  similar,  indicate  the  operation 
by  the  mintis  sign. 

EXAMPLES. 

1.  What  is  the  difference  between  Say/b  and  ay^? 
Here,  Sa-y/S  —  ay^  =  2ay/b,    Ans, 

145.  Give  the  nile  for  the  subtraction  of  mdlcals. 


190 


ELEMENT  All  Y       ALGEBRA. 


2.  From   9a -^2^  subtract   6a ^27^. 
First,     9av^762  =  27a5'v/3,   and  ^a^/¥iW'  =  lSab\/3; 
and,  27  ab^  —  ISab^  =  9ab\/3.    Ans, 

Find  the  differences  between  the  following : 


3.  ^15   and   -v/iS- 

4.  -v/24a252  and    \/E4b\ 


Ans.  {2ab  —  db^)y^. 


6.  y/l2SaW  and    'v/32a9. 

7.  v^48a3^  and    Vdab. 


Ans.  —Vis. 
45 

Ans.  {8ab  —  4a*)  ^20] 

-4ws.  4abV3ab  —  S\/ab. 


8.  V242a52,5  and    ^2a^K     Ans,  {Ua^^  —  ab)\/2ab. 


^-  \/l  ^^^  vl 


10.  V'320a2   and   ^80a\ 


Ans.  --/3. 
6 

Ans,  4a  y^. 

11.  ^^200^   and   ^24.5abc^d\ 

Ans,  {12ab  —  Ycc?)-/5a6. 

32.  -v/oeSo^^   and    v^OOo^^.  u4w«.  12aJv^ 

13.  ■v/ll2a^   and    '/28a86«.  Ans.  2a^b^y/V, 


MULTIPLICATION    OF    RADICALS. 

E46.     Radicals  are  multiplied  like  other  algebraic  quan* 
litics ;  hence,  we  have  the  following 

BULE. 

I.  Multiply  the  coefficients  together  for  a  new  coefficient: 


146.  Give  the  rule  for  the  multiplication  of  radicals. 


DIVISION      OF      EADI0AL9.  191 

n.  Multiply  togetJier  the  quantities  under  the  radical 
signs  : 
in.  Tlien  reduce  the  result  to  its  simplest  form, 

1.  Multiply  3a V^  hj  2\/ab, 

da\/bc  X  2^0^  =  3a  X  2  X  y^  X   V^' 
which,  by  Art.  139,  =  Qa^/b^ac  =  Qab-^, 

Multiply  the  following : 

2.  S\/5ab  and  4v^0a.  Ans.  120a  V^, 
8.  2aV?c  and  3aV?c.  Ans,  6ii^bc, 

4.  2a-v/aM^  and   -  Sa^a^+b\  A.  -  6a'(a'^  +  b\) 

5.  2a5-/a  +  ^>  and  ocy^a  —  b.     Ans.  la^c^a^  —  l^. 

6.  ny^  and   2-v/8.  Ans.  24. 
^    7.  \y/lc^  and   y»y-/|^.                       ^n5.  ^JyaJcyTs; 

8.  2aj  +  v^  and   2a;  —  y^.  Ans,  Aa?  —  h 

9.  \/a  4-  2-/b  and  -v^a  —  2-v/6.  -4?i5.  y^a^  —  45. 
10.  3aV^7a3  by    y^.  -4w«.  9a3y/6" 


DIVISION     OP     RADICALS. 


147.     Radical  quantities  are  divided  like  other  algebraic 
quantities ;  hence,  we  have  the  following 


BULB. 


L  Divide  the  coefficient  of  the  dividend  by  the  coefficient 
of  the.  divisor^  for  a  new  coefficient : 


147.  Give  the  rule  for  the  division  of  radicals. 


192  ELEMENTARY       ALGEBRA. 

II.   Divide  tJie  quantities  under  the  radicals,  in  tJie  same 
manner  : 
m.    The7i  reduce  the  result  to  its  simplest  form, 

EXAMPLES. 

1.  Divide   8a  y^  by  4a  y^. 

8a        ^  «... 

■—  =  2,    new  coemcient. 
4a 

hence,  the  quotient  is     2    X  -  =  • 

c  c 

2.  Di\ide    5a\/h    by    2by/c.  Ans.  -^v/-- 

3.  Divide    12ac-}/6bc    by    4c y^.  Ans.   3a ^3^. 

4.  Divide    6a-v/966*    by    SyW.  ^?i5.  4aJV^ 

5.  Di\ide    4a^^/50b^    by    2a2y^.  ^ns.   2b^^^, 


6.  Divide  26a^b\/81aFb^  by    13av/9a^.  A.  ea'^b^/ab. 

1,  Divide  84a35*-/27ac    by    42a5-/3a.  ^.  ea^j^-y/c. 

8.  Divide  ^/\c^   by    V^.  ^ws.   la. 

9.  Divide  Qa^b'^^/20a^    by    12-v/5a.  Ans.   a^b\ 

10.  Divide  6a -v/lO^    by    Z^/b,  Ans,  2ab^, 

11.  Divide  486*  y^    by    25^-/^  ^m.   Zmb\ 

12.  Divide  8a25*c3V^    by    2a^2M.  Ans.   2a¥cH, 

13.  Divide  OGa^c^y'gS^    by    48a5cV^.  -4.    Ua^^ic^. 


t^QUARE      BOOT      OF      POLYNOMIALS.  Wd 

14.  Divide    27a«i«-|/2ia3    by    -/Ta-        ^'*^-   27a«6°V5. 
16.  Divile    ISa^^VSo*    by    Qaby/a\      Am.   Qa^b^-y^. 

SQUARE     ROOl      OF     POLYNOMLA.LS. 

I4H.  Before  explaiiimg  the  rule  for  the  extraction  of  the 
bqiiare  root  of  a  polynoinial,  let  us  first  examine  the  squares 
of  Beveral  polynomials :  we  have, 

(a  +  by  =  a^  +  2rt6  4-  b\ 

(a  +  b  -^  cy  =  a^  +  2ab  +  b^  +  2{a  -f-  b)c  -}-  c», 

{a  -h  b  +  c  -{-  dy  =  a^  +  2ab  -\-  b^ -\-  2(a  •}-  b)c  -\-  cr" 

H-  2(a  4-  *  +  c)d  -r  c^. 
The  law  by  which  these  squares  are  formed  can  be  enim 
ciated  thus : 

The  square  of  any  polynomial  is  equal  to  the  square  of 
tfie  Jirst  term^  plus  twice  the  j/roduct  of  the  first  tenn  by  the 
second^  plus  tlie  square  of  the  second ;  plus  twice  the  first 
two  terms  midtipUed  by  the  thirds  plus  the  square  of  the 
third ;  plus  twice  t/ie  first  three  terms  7mdtiplled  by  the 
fourth,  plus  the  square  of  the  fourth;  aiid  so  on. 

149.     Hence,  to  extract  the  square  root  of  a  pol}^lomial, 

w  L'  have  the  following 

BULB. 

L  Arrange  the  polynomial  with  reference  to  mve  of  its 
letters^  arid  extract  tfie  square  root  of  the  first  term :  this 
will  give  the  first  term  of  the  root : 

148.  What  is  the  square  o^ a  binomial  equal  to?  What  is  the  sqnaro 
of  a  trinomial  equal  to?     To  what  is  the  square  of  any  polynomial  equal? 

149.  Give  the  rule  for  extracting  the  mjunre  root  of  a  polynomial? 
Wuat  is  the  first  step*  Wl  at  the  second  ?  WiuM  the  third  ?  What  the 
fourth  ? 

9 


IM  ELEMENTARY       ALQEBKA. 

n.  Divide  the  second  term  of  the  polynorrdal  by  double, 
the  first  term  of  the  rovt^  and  the  quotient  will  be  the  second 
term  of  the  root  : 

nL  Then  form  the  square  of  the  algebraic  sum  of  the 
two  terms  of  the  root  founds  and  subtract  it  from  the  first 
polynomials  and  then  divide  the  first  term  of  the  remainder 
hy  double  the  first  term  of  the  root^  and  the  quotient  will  be 
the  third  term : 

rV.  Form  the  doid)le  product  of  the  sum  of  the  first  a?ul 
second  terms  by  the  third,  and  add  the  square  of  the  third  ; 
then  subtract  this  residt  from  the  last  remainder,  and  divide 
the  first  term  of  the  result  so  obtained,  by  double  the  first 
term  of  the  root,  and  the  quotient  will  be  the  fourth  term. 
Then  proceed  in  a  similar  manner  tofitid  the  other  terms. 

EXAMPLES. 

1.  Extract  the  square  root  of  the  polynomial, 

49^2^2  _  24aZ>3  -f  25a*  -  ^Od'b  +  16Z>*. 
First  arrange  it  with  reference  to  the  letter  a. 


25a*  - 
25a*- 

-  30a^6  +  49a2^>2  _ 

-  30a^5  +     Qd'b'^ 

-  24ab^  +  IQb^ 

5a2  _  Sab  4-  4b^ 
10a2 

40a^b^  - 
40a'62  - 

-  24a53  +  IGJ* 

-  2-iab^  +  16b^ 

0     .     .     .     . 

.     .     1st  Hem, 
.     .     2d  Bem. 

After  having  arranged  the  polynomial  with  reference  to 
a,  extract  the  square  root  of  25a* ;  this  gives  ba^,  which 
is  placed  at  the  right  of  the  polynomial :  then  divide  the 
second  terra,  —  SOa^S,  by  the  double  of  bd^,  or  10a-; 
the  quotient  is  —  ^ab,  which  is  placed  at  the  right  of  ba^. 
H(mce,  the  first  two  terms  of  t*ie  root  are  ba^  —  3ab. 
Squaring  this  binomial,  it  becomes  25a*  —  SOd^b  +  9a'^b% 
which,  subtracted  from  the  proposed  polynomial,  gives  a 
remainder,  of  which  the  first  terra  is   40a,'^b\     Dividing  this 


8QDABB      ROOT      OP      P  O  L  I  N  ;>  M  I  A  t  B  .         195 


first  term  by  10a  ,  (the  double  of  Sa^),  the  quotient  ia 
■f  4ft2 ;  this  is  the  third  term  of  the  root,  and  is  written  on 
the  right  of  the  first  two  terms.  By  forming  the  double 
product  of  5a^  —  Sad  by  4^^^  squaring  ib\  and  taking 
the  sura,  we  find  the  polynomial  40a'^b^  —  24ab^  -\-  106% 
which,  subtracted  from  the  first  remainder,  gives  0.  There- 
fore, ba^  —  3ab  +  4b^   is  the  requii*ed  root. 

2.  Find  the  square  root  of  a*+  4a^x-\-Qa^x^+Aaa:^-{-  jb*' 

Ans.   a^-h  2aa;  -h  x^, 

8.  Fmd  the  square  root  of   a*—  Aa^x-^-Qa^x"^—  4ax^-\-  x\ 

Ans,  a}  —  2ax  +  sc*. 
4:  Find  the  square  root  of 

4x^  +  12x*  -\-  5x*  -  2x^+  1x^  -  2a;  +  1. 

Ans,   2x^  -\-  dx^  —  X  -h  1, 

5.  Find  the  square  root  of 

9a*  -  I2a^b  +  SSa^J'  -  IQab^  +  166*. 

Ans,   3a2  _  2ab  +  4^2. 

6.  Wliat  is  the  square  root  of 

«♦  —  4ax^  +  4a2a;2  —  4x^  +  Sax  -\-  4? 

Ans.   x^  —  2ax  —  2. 

7.  What  is  the  square  root  of 

9a;2  —  12a;  +  Gary  f  y^  —  4y  +  4  ? 

A71S,  Sx  f  y  —  2, 

8.  What  is  the  square  root  of  y*  —  2y^ar*  4-  2a;2  _  2y» 
+  1  +  a:*  ?  Ans.  if~9?^i, 

9.  What  is  the  square  root  of   9a^6*  —  30a^63+  25a^b^? 

Ans.   Za^b^  —  5ab, 
10.  Find  the  square  root  of 
25a*A*  -  40a362o  -r  16a^^c^  -  4Sab^c^  f  3C6V  —  SOa^bc 

-f  24a36c*  -  SQa^bc^  -f  9a*c2.  X 

A?is.   ba^b  -  Za'^c  —  4abc  +  06o^ 


196  ELEMENT  AKY       ALGEBKA. 

150.     W3  will  conclude  this  subject  with  the  following 
remarks : 

1st.  A  binomial  can  never  be  a  perfect  square,  since  we 
know  that  the  square  of  the  most  simple  j)olynomial,  viz. 
a  binomial,  contains  three  distinct  parts,  which  cannot  ex 
perience  any  reduction  amongst  themselves.  Thus,  the 
exj)ressi()n  a^  -\-  b"^,  is  not  a  perfect  square ;  it  wants  the 
term   ±  2aby  in  order  that  it  should  be  the  square  of  a  ±  6. 

2d.  In  order  that  a  trinomial,  when  arranged,  may  be  a 

perfect  square,  its  two  extreme  terms  must  be  squares,  and 

the  middle  term  must  be  the  double  product  of  the  square 

roots  of  the  two  others.    Therefore,  to  obtain  the  square 

root  of  a  trinomial  when  it  is  a  perfect  square :     Extract  the 

roots  of  the  two  extreme  terms,  and  give  these  roots  the  sams 

or  contrary  signs,  according  as  the  middle  term  is  positive 

or  negative.     To  verify  it,  see  if  the  double  product  of  the 

two  roots  is  the  same  as  the  middle  term  of  the  triJiomiaU 

Thus, 

9a^  —  48^4^2  j^  64a^^*,    is  a  perfect  square, 


smce,        v^So^  =  Sa^,    and    -/e^o^^^  =   —  Sad^ ; 

and  also, 

2  X  3a3  X  —  8a52  =   —  48a*^>2  =  the  middle  term. 

But,  4a2  _}-  14^5  +  9Z>2  is  not  a  perfect  square :  for, 
although  4^2  and  +  96^  are  the  squares  of  2a  and  36, 
yet    2  X  2a  X  36    is  not  equal  to    14a6. 

3d.  In  the  series  of  operations  required  by  the  general 
rule,  when  the  first  term  of  one  of  th^  remainders  is  not 
exactly  divisible  by  twice  the  first  term  of  the  root,  we  may 

160.  Can  a  binomial  ever  be  a  perfect  power?  Why  not?  When  is 
h  trinomial  a  perfect  square  ?  When,  in  extracting  the  square  root,  wc 
find  that  the  first  term  of  the  remainder  is  not  divisible  by  twice  the  root, 
ifl  the  polynomial  a  perfect  power  or  not? 


SQUARE   ROOT   OF   POLYNOMIALS.    197 

oonclude  that  the  proposed  polynomial  is  not  a  perfect 
square.  Tliis  is  an  evident  consequence  of  the  course  of 
reasoning  by  winch  vre  have  arrived  at  the  general  rule  fo? 
extracting  the  square  root. 

4ih.  When  the  polynomial  is  lot  a  perfect  square,  it  may 
sometimes  be  simplilied  (See  Ait.  139). 


Take,  for  example,  the  expression,  -^a^b  4-  4a^b^  -f  4ab^ 

The  quantity  under  the  radical  is  not  a  perfect  square ; 
but  it  can  be  put  under  the  form  a5(a^  +  4ab  -h  4h^.) 
Now,  the  factor  within  the  parenthesis  is  evidently  the 
square  of  a  +  26,    whence,  we  may  conclude  that, 


\^a^b  +  4aV)^  +  4<(b'  =   (a  +  2b)  y/ab, 

2.  Reduce    -/2a»6  —  4ab'^  -\-  2b^    to  its  simplest  form. 

Ans,   (a  —  h)  -v/SS 


198  ELEMENTARY      ALGEBRA 


CHATTER    Yin, 

BQUATIONS       OP       THE       SECOND        DEGREE. 
EQUATIONS    CONTAINING    ONE    UNKNOWN    QUANTITY. 

151.  An  Equation  of  the  second  degree  containing  but 
one  unknown  quantity,  is  one  in  which  the  greatest  exponent 
is  equal  to  2.     Thus,  ' 

x^  =z  a,       ax^  -\-  bx  =  c, 

are  equations  of  the  second  degree. 

152.  Let  us  see  to  what  form  every  equation  of  the 
second  degree  may  be  reduced. 

Take  any  equation  of  the  second  degree,  as. 

Clearing  of  fractions,  and  perforirdng  indicated  operations, 
Nve  have, 

4  -f  805  -h  4a;2  —  3a;  —  40  =  20  —  «  +  2x\ 

Transposing  the  unknown  terms  to  the  first  member,  the 
kno\^Ti  terms  to  the  second,  and  arrangmg  with  reference  to 
the  powers  of  a?,  we  hftve, 

4a;2  —  2a;2  +  Sic  -  3a;  +  a;  rr:  20  +  40  —  4 ; 

151.  What  is  an  equation  of  the  second  degree?     Give  an  example. 

152.  To  what  form  may  every  equation  of  the  second  degree  be  reduced? 


EQUATIONS      OF     THE     SECOND      DEGREE.       199 

and,  by  reducing, 

2x^  +  Qx  =  56 ; 

dividing  by  the  coefficient  of  jc^,  we  have, 

a;2  +  3a;  =  28 

If  we  denote  the  coefficient  of  x  by  2;>,  and  the  second 
member  by  y,  we  have, 

a*  +  SjtxB  =  q. 

Tins  is  called  the  reduced  equation. 

153.  When  the  reduced  equation  is  of  this  form,  it  con- 
tains three  terms,  and  is  called  a  complete  equation.  The 
terms  are, 

First  Term. — ^Tlie  second  power  of  the  unknown  quan- 
tity, vnih  a  plus  sign. 

Second  Term. — ^The  first  power  of  the  unknown  quantity, 
with  a  coefficient. 

Third  Term. — A  known  term,  in  the  second  member. 

Every  equation  of  the  second  degree  may  be  reduced  to 
this  form,  by  the  following 

rule. 

L  Ch"r  tJii'  equation  of  fractions^  and  perform  all  the 
indicated  operations  : 

n.  Tran,spose  all  theunhw<r/t  t< nns  to  tJie  first  mewher^ 
and  all  t/ie  k?ioton  terms  to  t/ie  second  member  : 

158.  How  many  tcnns  are  thcrtj  In  a  complete  equation  ?  What  is  the 
flrpt  tcrtu  ?  What  13  the  second  terra  ?  What  is  the  third  terra?  How 
many  oj^rationa  are  there  iu  reducing  an  equation  of  the  second  depr^n* 
to  the  required  form  ?  What  b  the  first  ?  What  the  second  ?  What  the 
third  ?     What  the  fourth  ? 


200  ELEMENTARY       ALGEBRA 

m.  Redxice  all  the  terms  containing  the  square  of  the 
unknown  quantity  to  a  single  term^  07ie  factor  of  which  is 
the  square  of  the  unknown  quantity  ;  reduce^  also^  all  the 
terms  containing  the  first  power  of  the  unknoicn  quantity ^ 
to  a  single  term  : 

rV.  Divide  both  members  of  the  resulting  equation  hy 
the  coefficient  of  the  square  of  the  unknotcn  quantity. 

154.  A  Root  of  an  equation  is  such  a  value  of  the  un- 
known quantity  as,  being  substituted  for  it,  will  satisfy  the 
equation  ;  that  is,  make  the  two  members  equal. 

The  Solution  of  an  eijuation  is  the  operation  of  finding 
its  roots. 

INCOMPLETE    EQUATIONS. 

155.  It  may  happen,  that  2/),  the  coefficient  of  the  first 
power  of  X,  in  the  equation  x  +  2px  =  q^  is  equal  to  0. 
In  this  case,  the  first  power  of  x  will  disappear,  and  the 
equation  will  take  the  form, 

^=  q (1-) 

This  is  called  an  incomplete  equation  ;  hence, 

An  INCOMPLETE  EQUATION,  whcn  rcduccd,  contains  but 
two  terms ;  the  square  of  the  unknown  quantity,  and  a 
known  term. 

156.  Extracting  the  square  root  of  both  members  of 
Equation  ( 1 ),  we  have, 

X  z=    ±^/q. 

1 54.  What  is  the  root  of  an  equation  ?  What  is  the  solution  of  an 
equation  ? 

155.  What  form  will  the  reduced  equation  take  when  the  coefficient  ot 
ar  is  0  ?  What  is  the  equation  then  called  ?  How  many  terras  are  there 
hi  an  incomplete  equatiou  ?     What  are  they  ? 

156.  What  is  the  rule  for  the  solution  of  an  incomplete  equation? 
How  many  roots  are  there  in  every  incomplete  eqratiou  ?  How  ao  thr 
roots  ooinpai  ^  v\  ith  each  other  ? 


EQUATIONS     OF     THE     SECOND     DEQEKE.       201 

Hence,  for  the  solution  of  incomplete  equations: 

R  u  L  K. 

I.   Heduce  the  equation  to  the  form  a?  •=:  q  : 
II     The7i  extract  the  square  root  of  both  members. 

Note. — ^There  "will  be  two  roots,  numerically  equal,  but 
having  contrary  signs.  Denoting  the  first  by  x\  and  the 
second  by  «",  we  have, 

jc'  =   4-  y^,     and     cc"  =   —  y^. 

verification; 

Substituting  -f-  y^,  or  —  y^,  for  jc,  in  Equation  ( 1 ), 
we  have, 

{+V7jY  =  q\     and,     (-Vv)'=  r. 

h^nce,  both  satisfy  the  equation  ;  they  are,  therefore,  roots 
(Art.  154.) 

EXAifPLES. 

1.  What  are  the  values  of  x  in  the  equation, 

3^2^  8  _  5^2-10? 

By  transposing,    .Sx^  —  hoi?-  —   —  1 0  —  8. 

Reducing,  —  It?  ^   —  IB.    • 

Dividing  by  —  2,  t?  —  ^. 

Extracting  square  root,     a:  =    rfc  \/9  =4-3   and    —  y. 

Hence,  a?'  =   4-  3,   and  ar"  =   —  3, 

2.  What  are  the  roots  of  the  equation, 

3a^  4-  6  =  4a;2  -  10? 

Am.   x'  =    +4,    x"  =   -  1. 


202  ELEMENTARir      ALGEBRA. 

3.  What  are  the  roots  of  the  equation, 

A?is.   x'  =   +9,    a"  =   —  9, 

4.  What  are  the  roots  of  the  equation, 

4a;2+  13  —  2a;2  =  45  ? 

A?is.   x'  =   +  4,    a"  =   —  4. 

6.  What  are  the  roots  of  the  equation, 
Qx^  —  1  =:  3a;2+  5? 

Ans.   x'  =   +2,    aj"  =   —  2. 

6.  What  are  the  roots  of  the  equation, 

8  +  5a;2  =  ^  _|-  4a;2 -f  28  ? 
6 

Ans.  x'  =z   -1-5,    £b"  =   —  5. 
V.  What  are  the  roots  of  the  equation, 

8  3 

Ans.   a'  =   +6,    aj"  =   —  5. 

8.  What  are  the  roots  of  the  equation, 
x"^  -\-  ab  -  5x^  ? 
Ans.  a'  =   -f-  Jy^,    a"  =  -  Jy^ 
9    What  are  the  roots  of  the  equation. 


xy/a  +  a;2  =  b  -j-  x^? 

Ans.   x'  —  ,    a;"  =   —  — 

V«  —  2b  ^/a  —  2b 


PROELEMS. 


FBaBLEMS. 


208 


1.  What  number  is  that  which  being  multiplied  by  itself 
the  product  will  be  144  ? 

Let  X  =  the  number :  then, 

X  X  X  =  x^  —  144. 

It  is  plain  that  the  value  of  x  will  be  found  by  extracting 
the  square  root  of  both  nlerabers  of  the  equation ;  that  is, 

■V^  =  ylii :  that  is,    a;  =  12. 

2.  A  person  being  asked  how  much  money  he  had,  said, 
if  the  number  of  dollars  be  squared  and  6  be  added,  the  wmi 
wiW  he  42 :  how  much  had  he  ? 

Let  X  =   the  number  of  dollars. 

Tlien,  by  the  conditions, 

a*  4-  6   =  42 ; 
hence,  aj^  _  42  _  o  =  36, 

and,  jB  =     6.  Arts.   |6, 

3.  A  grocer  being  asked  how  much  sugar  he  had  sold  to 
a  person,  answered,  if  the  square  of  the  number  of  pounds 
be  multiplied  by  7,  the  product  will  be  1575.  How  many 
pounds  had  he  sold  ? 

Denote  tlie  number  of  pounds  by  x.  Then,  by  the  con- 
ditions  of  the  question,^ 

1a^  =  1575  ; 

hence,  x"  =    225, 

and,  X    =z       15.  Anji.    15. 

4.  A  person  being  asked  his  age,  said,  if  from  the  square 


204  ELEMENTARY        ALGEBRA. 

of  ray  age  in  years,  you  take  1 92  years,  the  remainder  will 
be  the  square  of  half  my  age :  what  was  his  age  ? 

Denote  the  number  of  years  in  his  age  by  as. 

Then,  by  the  conditions  of  the  question, 

aj2 


192 


=  ii-} 


and  b)'  clearing  the  fractions, 

4ic2  _  768   =  a;2 ; 

hence,  ix^  —  x^  =  768^ 

and,  3a*2  =  7(38, 

x^  -  256 

X    =  16  Ans.    16  years. 

5.  What  number  is  that  whose  eiglitn  part  multiplied  by 
its  fifth  part  and  the  product  divided  by  4,  will  give  a  quo- 
tient equal  to  40  ? 

Let   a;  =:  the  number. 

By  the  conditions  of  the  question, 


(V  ""V)-^*  ^  '•"' 


by  clearing  of  fractions, 

x^  z=z   6400, 

X    =       80.  Ans.    80. 

6.  Find  a  number  such  that  one-third  of  it  multiplied  by 
one  fourth 'fehall  be  equal  to  108.  Ans.    36. 

7.  Wliat  number  is  that  whose  sixth  part  multiplied  by 
its  fifth  part  and  the  product  divided  by  ten,  will  give  a 
qiiotii'nt  equal  to  3  ?  A7is.    SO 


PROBLEMS.  205 

8.  What  number  is  that  who8e  square,  phis  18,  will  be 
equal  to  half  the  square,  plus  30^  ?  A/ts.    5. 

9.  What  numbers  are  those  which  are  to  each  otlier  as 
1  to  2,  and  the  diflerence  of  whose  squares  is  equal  to  75  ? 

Let     X  =  the  less  number. 
Then,    2x  =  the  greater. 

Tlien,  by  the  conditions  of  the  question, 

4iB»  -  jb2  =   75  ; 

hence,  3a^  =  75, 

and  by  dividing  by  3,    s^  =  25,    and    jb  =  5, 

and,  2x  =  10. 

An8.   5  and  10 

10.  What  two  numbers  are  those  which  are  to  each  othcf 
as  5  to  6,  and  the  difference  of  whose  squares  is  44  ? 

Let     X  =  the  greater  number. 

Then,   -x  =  the  less. 

By  the  conditions  of  the  problem, 

«2  -  ~x^  =  44  ; 
30  * 

by  clearing  of  fractions, 

30a:*  -  25x2  =  1684  ; 

hence,                                  lla^  =  1584, 

and,                                           a^  =  144  ; 

hence,  X    =       12, 

5 
and,  A^    ~       ^^' 

A718.    10  and  12 


206  ELKMENTAKY       xLQEBRA. 

11.  "Wnat  two  D umbers  are  those  which  are  to  each  other 
ai5  3  to  4,  and  the  difference  of  whose  squares  is  28  ? 

A71S.   6  and  8. 

12.  What  two  numbers  are  those  which  are  to  each  other 
as  5  to  11,  and  the  sum  of  whose  squares  is  584  ? 

Ans.    10  and  22. 

13.  ^  says  to  JB,  my  son's  age  is  one  quarter  of  yours, 
and  the  difference  between  the  squares  of  the  numbers 
representing  their  ages  is  240  :  what  were  their  ages  ? 

Eldest,       16, 


(  Eldest,       16, 

Ans.     ]  ^^  ' 

(  lounger,     4. 


Two  u7iknoio7i  quantities. 
157.     When  there  are  two  or  more  unknown  quantities: 

I.  JElimiiiate  one  of  the  unknown  quantities  hy  Art. 
113.- 

■   n.  Then  extract  the  square  root  of  both  members  of  the 
equation. 

PROBLEMS. 

1.  There  is  a  room  of  such  dimensions,  that  the  difference 
of  the  sides  multiplied  by  the  less,  is  equal  to  36,  and  the 
product  of  the  sides  is  equal  to  360  :  what  are  the  sides  ? 

Let  X  =   the  length  of  the  less  side  ; 

2/  =    the  length  of  the  greater. 
Then,  by  the  first  condition, 

(y  —  x)x  =     36  ; 
and  by  the  2d,  xi/  =  360. 

157.  How  do  yoa  proceed  when  there  are  two  or  more  unknown  quan- 
litifcfl  ? 


PROBLEMS.  207 

From  the  first  eqaation,  we  have, 

ay  —  a;2  _     qq  . 

and  by  subtraction,  x^  =  324. 

Hence,  x  =  y's^  =18; 

V  = =     20. 

^  18 

Ans,  aj  =  18,    y  =  20. 

2.  A  merchant  sells  two  pieces  of  muslin,  which  together 
measure  12  yards.  He  received  for  each  piece  just  so  many 
dollars  per  yard  as  the  piece  contained  yards.  Now,  he  gets 
four  tmies  as  much  for  one  piece  as  for  the  oilier :  how  many 
yards  in  each  piece  ? 

Let    X  =   the  number  of  yards  in  the  larger  piece ; 

y  =   the  number  of  yards  in  the  shorter  piece. 

Then,  by  the  conditions  of  the  question, 

35  +  y  =  12. 

X  X  X  =  x^  =  what  he  got  for  the  larger  piece ; 

y  X  y  =  y^  =  what  he  got  for  the  shorter; 

and,  x^  =  4y\  by  the  2d  condition, 

x   =  2y,    by  extractmg  the  squai-e  root. 

Substituting  this  value  of  a;  in  the  first  equation,  we  have, 

y  +  2y  =  12; 

and,  consequently,  y  =  4, 

and,  a;  =  8. 

Ans.  8  and  4. 

8.  Wliat  two  numbers  are  those  whose  product  is  30,  and 
tlie  quotient  of  the  greater  by  the  less,  3^  ?    Ans.  10  and  3. 

4.  The  product  of  two  numbers  is  a,  and  then*  quotient 

6:  what  are  the  numbers?  /^ 

Ans.   \/uJ>y   and  v/^  . 


208  ELEMENTARY       ALGEBRA. 

6.  The  sum  of  the  squares  of  two  numbers  is  117,  and  the 
difference  of  their  squares  45  :  what  are  the  numbers  ? 

A?7S.  9  and  6. 

6.  The  sum  of  the  squares  of  two  numbers  is  a,  and  the 
difference  of  their  squares  is  b  :  what  are  the  numbers  ? 


.  fa  -^  b  fa 


1.  What  two  numbers  are  those  which  are  to  each  other 
as  3  to  4,  and  the  sum  of  whose  squares  is  225  ? 

Ans.  9  and  12 

8.  Wliat  two  numbers  are  those  which  are  to  each  other 
as  m  to  n,  and  the  sum  of  whose  squares  is  equal  to  a^  ? 

ma  na 

9.  Wliat  two  numbers  are  those  which  are  to  each  other 
as  1  to  2,  and  the  difference  of  whose  squares  is  75  ? 

Ans.  5  and  10. 

10.  What  two  numbers  are  those  which  are  to  each  other 
as  m  to  n^  and  the  difference  of  whose  squares  is  equal  to  b"^  ? 

mb  nb 


Ans. 


^/wfi  —  n^      'y/n 


11.  A  certain  sum  of  money  is  placed  at  interest  for  six 
months,  at  8  per  cent,  per  annum.  Now,  if  the  sum  put  fit 
interest  be  multiphed  by  the  number  expressing  the  interest, 
the  product  will  be  $562500  :  what  is  the  principal  at  m- 
tc^rest?  Ans.  $3750. 

12.  A  person  distributes  a  sum  of  money  between  a  num- 
ber oi  women  and  boys.  The  number  of  women  is  to  the 
number  of  boys  as  3  to  4.  Now,  the  boys  receive  one-half 
as  many  dollars  as  there  are  persons,  and  the  women,  t^^ce 
as  many  dollars  as  there  are  boys,  and  together  they  receive 


COMPLETE      EQUATIONS.  209 

138  dollars :  how  many  women  were  there,  and  how  many 
boys? 

.       j  36  women. 
(  48  boys. 

COMPLETE    EQUATIONS. 

t^H  The  reduced  form  of  the  complete  equation  (Art, 
168)  is, 

Comparing  the  first  member  of  this  equation  with  the 
pquare  of  a  binomial  (Art.  54),  we  see  that  it  needs  but  the 
square  of  half  the  coefficient  of  a;,  to  make  it  a  perfect  square. 
Adding  p^  to  both  members  (Ax.  1,  Art.  102)^  we  have, 

x^  +  2px  -\-  p^  z=z  q  ■\-  p\ 

Then,  extractmg  the  square  root  of  both  members  (Ax.  5), 
we  have,  

X  -\-  p    z=z     ±  'v/^~T^. 

Transposing  p  to  the  second  member,  we  have. 


X  =   —  p  ±  vY-1- P^' 

Hence,  there  are  two  roots,  one  corresponding  to  the  2^U8 
sign  of  the  radical,  and  the  other  to  the  minus  sign.  De- 
noting these  roots  by  a'  and  a",  we  have, 


cb'  =   —  /)  +  \q-\-  p\    and    a"  =   —  p  —  ^/q  -\-  p^. 

The  root  denoted  by  x'  is  called  the  first  root ;  that  de- 
noted by  x'*  is  called  the  second  root, 

168.  What  is  the  form  of  the  reduced  equation  of  the  second  <legree? 
Wliat  is  the  square  of  the  binomial  x  +  p  ?  How  many  of  those  tcrnig 
are  found  in  the  firsi  term  of  the  reduced  equation?  What  must  be 
added  to  make  tlie  first  member  a  perfect  square  ?  How  many  roots  are 
there  in  every  equation  of  the  first  degree!  What  is  the  first  root  equaJ 
to  ?     What  is  tlie  second  equal  to  7 


210  ELEMENTARY      ALGEBRA 

159.  The  operation  of  squaring  half  the  coefficient  of 
X  and  adduig  the  result  to  both  members  of  the  equation,  ia 
called  Gomj^letiiig  the  Square.  For  the  solution  of  every 
complete  equation  of  the  second  degree,  we  have  the  foj- 
lowmg 

RULE. 

L   Reduce  the  equation  to  theform^  x^  +  2^93;  z=  q: 

11.  Take  half  the  coefficient  of  the  second  term.,  square 
it,  and  add  the  result  to  both  members  of  the  equation  : 

in.  Then  extract  the  square  root  of  both  members  ;  after 
which,  transpose  the  hnowyi  term  to  the  second  member. 

Note. — Although,  in  the  beginning,  the  student  should 
complete  the  square  and  then  extract  the  square  root,  yet 
he  should  be  able,  in  all  cases,  to  write  the  roots  immediately, 
by  the  following     (See  Art.  158) 

RULE. 

I.  The  first  root  is  equal  to  half  the  coefficient  of  the 
second  term  of  the  reduced  equation,  taken  with  a  contrary 
sign,  plus  the  square  root  of  the  second  member  iiicreased 
by  the  square  of  half  the  coefficient  of  the  second  term  : 

n.  The  seco7id  root  is  equal  to  half  the  coefficient  of  the 
second  term  of  the  reduced  equatioji,  taken  with  a  contrary 
sign,  m^inus  the  square  root  of  the  second  member  increased 
by  the  square  of  half  the  coefficient  of  the  second  term. 

160.     We  will  now  show  that  the  comi)Iete  equation  of 

159.  What  is  the  operation  of  completing  the  square?  How  many 
operations  are  there  in  the  sohition  of  every  equation  of  the  second  de- 
gree ?  What  is  the  first  ?  What  the  second?  What  the  third ?  Give 
the  rule  for  writing  the  roots  without  completing  the  square? 

ISO.  How  many  forms  will  the  complete  equation  of  the  second  degree 
assume?     On  what  will  these  forms  depend?     What  are  the  signs  of  2/- 


OOMPLEIE      EQUATIONS.  211 

the  Becond  degree  will  take  four  foi-ms,  dependent  on  the 
signfi  of  1p  and  q. 

l8t.  Let  lis  suppose  2p  to  be  positive,  and  q  positive;  we 
shall  then  have, 

a;2+  27^  =  ^ (1.) 

2d.  Let  us  suppose  2/>  to  be  negative,  and  q  positi\'e 
we  shall  then  have, 

x^  —  2px  =  q (2.) 

3d,  Let  us  suppose  2/>  to  be  positive,  and  q  negative ; 
we  shall  then  have, 

jb2  4-  2;xc  =   -  5-.        .     .     .     ( 3.) 

4th.  Let  us  suppose  2p  to  be  negative,  and  q  negative ; 
we  shall  then  have, 

x^  —  2px  =   —  q.        ...     (4.) 

As  these  are  all  the  combinations  of  signs  that  can  take 
plao-e  between  2p  and  y,  we  conchide  that  every  complete 
equation  of  the  second  degree  will  be  reduced  to  one  or  the 
other  of  these  four  forms : 

a^  +  2px  =  +  q,  .  .  1st  form, 

aj*  —  2px  =  +  J',  .  .  2d    form. 

x^  +  2jyx  z=  ^  q,  ,  .  3d    form. 

x^  —  2/)ic  =  —  q^  .  .  4th  form. 

EXAMPLES    OF    THE    FIEST    FORM. 

1.  What  are  the  values  of  a;  in  the  equation, 

2a;2  +  8a;  =  64  ? 
If  we  first  divide  by  the  coefficient  2,  we  obtam 

a;2  4-  405  =  32. 

and  7  in  the  first  form?    What  in  the  second?    What  in  the  third? 
V-Tiat  in  the  fourth  ? 


212  ELEMENTARY       ALGEBEA. 

Then,  completing  the  square, 

a2  -f-  4a;  +  4   rzi   32  +  4   =  36. 

Extracting  the  root, 

a  +  2   =    ±  V  36   =    +  6,    and     —  6. 

Hence,  x'  =.    —  24-6   —    +4; 

and,  a"  =    —  2  —  6   =    —  8. 

rience,  in  this  form,  the  smaller  root,  numerically^  is  positive 
and  the  larger  negative. 

VERIFICATION. 

If  we  take  the  positive  value,  viz. :  a'  =   +4, 
the  equation,  a^  _^  4a.  _   32^ 

gives  42  -h  4  X  4  =   32  ; 

and  if  we  take  the  negative  value  of  jc,  viz. :   jc"  —   —  8, 
the  equation,  q(?  -\-  ^x  =:   32, 

gives  (—  8)2  +  4(—  8)   =   64  -  32   r=z   32; 

from   which  we  see  that  cither  of  the  values  of  a,    viz.j 
k'  =    +4,   or   x"  =   —  8,    will  satisfy  the  equation. 

2.  What  are  the  values  of  x  in  the  equation, 

3a;2  -f  12a;  —  19  =    —  x^  —  12a;  +  89? 
By  transposing  the  terms,  we  have, 

3a;2  4-  ^2  +  12a;  +  12a;  =   89  4-  19  ; 

and  by  reducing, 

4a;2  4-  24a;  =108; 

and  dividing  by  the  coefficient  of  x^^ 

aj2  4-  6a;  =   27. 


COMPLETE      EQUATIONS.  21b 

Now,  by  coinpletbig  the  square, 

ar^  +  6a;  -I-  9  =  36 ; 
extracting  the  Bquarc  root, 

a;  4-  3  =   ±  -/s'c.  =   +  6,   and    -  6: 
hence,  a:'  =   -f6  —  3   =   4-3; 

and,  a;"  =   —  6  —  3   =   —  9. 

VEUIFICATION. 

If  we  lake  the  plus  root,  the  equation, 

x^  +  Gx  =  27, 
gives  (3)'^+  0(3)   =  27; 

and  for  the  negative  root, 

a:2  -h  6a;  =  27, 
gives  (-  9)2  +  0(-  9)   =   81  -  64  =  27. 

3.  What  are  the  values  of  a;  in  the  equation, 

a;2  -  lOa;  -f  15   =    -  —  34a:  +  165  ? 
5 

By  clearing  effractions,  we  have, 

5^2  -  50a:  4-75  =  x^  —  170a;  4-  776; 
by  transposing  and  reducing,  we  obtain, 
4a^  4-  120a;  =700; 
then,  dividing  by  the  coefficient  of  a^y  we  have, 

a;2  4-    30a;  =   175; 
and  b}  completing  the  square. 


ar'  4-  30d;  4-  225   =  400 


*214         ELEMENTARY   ALGEBRA. 

and  by  extracting  the  square  root, 

a  -j-  15   =   iy'loo   =    +  20,   and    —  20. 
Hence,  x'  —   +5,    and   x"  —   —  35. 

VERIFICATION. 

For  the  plus  vahie  of  a;,  the  equation, 
a;2  _f-  30a;  =   175, 
gives,  (5)2  4-  30  X  5   =   25  +  150  =  176. 

And  for  the  negative  value  of  a;,  we  have, 

(_  35)2  ^  3o(_  35)    -   1225  —  1050   ~    175. 

4.  "VvTiat  are  the  vahies  of  a;  in  the  equation, 

Clearing  of  fractions,  we  have, 

10a;2  _  6a;  +  9  =  96  —  8a;  —  12a;2  -f  273; 

transposing  and  reducing, 

22a;2  -f-  2a;  =   360 ; 

di\dding  both  members  by  22, 

2       _   360 
"^    "^  22^  -     22  ' 

Add    I  —  )    to  both  members,  and  the  equation  becomes, 

o    .     2       .    /  1  \2        360        /  1  \2 
'^  +  22='+  (22)    =^  +  (25)' 

whence,  by  extracting  the  square  root, 


22  ~  ""  V  22  "^  \  '^2  / ' 


COMPLETE      EQUATIONS.  216 

therefore,  

.  1     ,       /3C0    .    /T\2 

^=    -2-2  +  V-22-  +  12-2)' 


and,  ."=  __l_,/!^+(iy^ 

22        V    22         V  22/ 

It  remains  to  perform  the  Dmnerical  opemtions.     In  the 
first  place, 


360 

22    "^ 


(y. 


mast  be  reduced  to  a  single  nmnber,  having  (22)^  for  its 
denominator.     Now, 


3G0        /  1   Y 
22         \22/    ~ 

3C0  X  22  4-  1          7921 

(22)2            -  (22)2' 

extracting  the  square  root  of  7021,  we  llnd  it  to  be  89 j 
therefore. 


/3G0     /  i~y  _     ?? 

V   22    "^  \22/    ~    "^  22* 


Consequently,  the  plus  value  of  x  is, 

*    ~         22        22   ~  22   ~     ' 
and  the  negative  value  is, 

"_    -  2   _  ??  —   _  1^. 
^    ~"         22        22   ~"    ~  11  ' 


that  is,  one  of  the  two  values  of  x  which  \»t11  stitisfy  the 
proposed  equation  is  a  positive  whole  number,  and  tlie  other 
a  negative  fraction. 

Note. — Let  the  pupil  be  exercised  in  writing  the  roots,  in 
the  last  five,  and  in  the  following  examples,  without  com- 
plcting  the  square. 


216         ELEMENT  AEY   ALGEBRA, 


6.  What  are  the  values  of  x  in  the  equation, 


j  JC'   =:    5. 

\x"-  -  r>% 


Ans. 


6    What  are  the  values  of  a;  in  the  equation, 
2a;2  +  8a;  +  7   =  ^  ~  ¥  "^  "^^^^ 

A71S.    i      „ 

(  x" 
7.  What  are  the  values  of  x  in  the  equation, 


-4    15   =  ^  -  8a;  4-  95i  ? 


=   8. 


-'''   {J- 


9. 
-64f 


8.  Wliat  are  tlie  values  of  x  in  the  equation, 

a;^        5a;  £c       ^         ^    o 

____8   =   --7a:  +  6i? 


(  x"=   -    7f 


0.  What  are  the  values  of  a;  in  the  equation, 


x^       x  _  x^        X         13 


A  j  a;'    =     1, 

Ans.   -j^,,^    _ 


% 


KXAMPI^S    OF  THE   SECOND   FORM. 

I.  What  are  tbe  values  of  x  in  the  equation, 
a;2  _  8a;  +  10   =:   19? 


COMTLKTE      EQUATIONS.  217 

By  transposing, 

aj2  -  8a;  =  19  -  10  =     0; 

then,  by  completing  the  square, 

jc2  _  8aj  -f  16  =     9  -h  10   =  25 ; 

and  by  eicti-acting  the  root, 

a;  -  4  =   ±  V^  =   +  5,    or    -6. 

Hence, 

a'  =  4  +  5  =  9,    and    a"  =  4  —  6  =.   -  1. 

That  is,  in  this  form,  the  larger  root,  numericallyj  ie 
positive,  and  the  lesser  negative. 

VERIFICATION. 

If  we  take  the  positive  value  of  cc,  the  equation, 

x^  —  8x  =  9,    gives     (9)^  —  8x9  =  81  —  72   =  9; 

and  if  we  take  the  negative  value,  the  equation, 

a^  -  8aj  =  9,    gives     (-  1)2  —  8  (—  1)  =  1  +  8  =  9; 

from  which  we  see  that  both  roots  alike  satisfy  the  equa- 
tion. 

2.  What  are  the  values  of  x  in  the  equation, 

X^  X  3*^ 

By  clearing  of  fractions,  we  have, 

6a;2  +  4a;  _-  leo  =  3x^  +  12aj  —  177  ,* 
and  by  transposing  and  reducing, 

Sx^  —  6x  z=  3  ; 
and  dividing  by  the  coefficient  of  cb*,  we  obtmn. 


218  ELEMENTAKY       ALGEBRA, 


Then,  by  completing  the  square,  we  have, 

2       8      ,16         ,    ,    16         25 
3^9  ^9  9  ' 

and  by  extracting  the  square  root, 

4         ^      /25  5-6 

a;  —  -=   ±\/  —  =   +-,  and  --  - 

3  V    9  3'  3 


Hence, 


«^=5  +  -3  =  +«'-^»"=S-i  =  -i 


VERIFICATION. 

For  the  positive  root  of  x,  the  equation, 

2  8 

x---x  =  1, 

Q 

gives  32  —  -x3   =  9  —  8  =  1 

3 

and  for  the  negative  root,  the  equation, 

2       8 
a'  -  3^  =  1, 

/      IV      8  1         1,8 

^^^^^  l-3J-3^-3  =  9  +  9  =  ^- 

8.  What  are  the  values  of  a;  in  the  equation, 

?-!  + Vf  =  8? 
2        3  * 

Cleanng  of  fractions,  and  dividing  by  the  coefficient  of 
x',  we  have, 

cc^  -  |c  =  H. 


COMPLETE      EQUATIONS.  219 

Completing  the  square,  we  have, 

2       2,1         ,,1         49 
^    -3^  +  9  =   '*  +  9  =  3a' 

then,  by  extracting  the  square  root,  we  have, 

X =   +\/  —  =   H —  ,    and    —  -  ; 

3         "^  V  36  6 '  .    6  * 

hence, 

^   =3  +  6   =  6  =  '^'    ^"^    ^    =3-6  =   -6' 


VERIFICATION. 

If  we  take  the  positive  root  of  x,  the  equation, 

gives  OiY-l  X  li  =  2i  -  1  =  li; 

and  for  the  negative  root,  the  equation, 

/      6V      2  6         25    .    10         45 

^''''      r  e)  -  3  ^  -  6-   =   -3l^  +  rs   =   36   =   ^' 

4.  What  are  the  values  of  JC  in  the  equation, 

4^2  _  2a;2  4-  2az  -   I8ab  —  186^? 

By  transposing,  changing  tlie  signs,  and  dividing  by  2, 
the  equation  becomes, 

x^  -  ax  =  2a'^  —  9ab  -^   9^ ; 


220  ELEMENTARY      ALGEBRA. 


whence,  completing  the  square, 
4  4 


extracting  the  square  root, 


a  /9a^ 


9a^ 
Now,  the  square  root  of   — ■  —  9ab  +  ^b'\    is  evidently 

-  ^  35.     Therefore, 

'"  =  2±(t-'*)'«°'^   ia="=-   a  +  Sb. 

What  will  be  the  numerical  values  of  x,  if  we  suppose 
a    =  6,   and    5  =  1? 

5.  What  are  the  values  of  a;  in  the  equation. 


1  4 

jB  —  4  -  a;2  4-  2a;  —  -a;2  =  45  —  3a;2  +  4a;  ? 


.        J  a;'    =        1.12  )  to  ^ 
^''''\x-=   -5.73f     0 


within 
01. 


6.  What  are  the  values  of  a;  in  the  equation, 

8a;2  —  14a;  +  10  =  2a;  +  34  ? 

7.  What  are  the  values  of  x  in  the  equation, 

J  -  30  +  a;  =  2a:  -  22  ? 

^"^-  { J'  Z  !i  1 

8.  Wliat  are  the  values  of  a;  in  the  equation, 
2 


a;2-  3a;  -f  ^  "^^  9a;  +  13^? 


Afis. 


\x'    =9, 
I  a-"  =  -  1. 


COMPLETE      EQUATIONS.  221 

9.  What  are  the  values  of  a;  in  the  equation, 
Caa;  —  a;2  =   —  2ah  —  ft^? 

(  x'    =  2a  -f  6. 
^""''ix^'^   -6. 

10.  What  are  the  values  of  x  m  the  equation, 

0»  +  52  __  2525  +  JT*   =    — Y-? 


Atis. 


EXAMPLES    OP    THE    THIRD    POEM. 

1.  What  are  the  values  of  a;  in  the  equation, 

a:2  +  4a;  =   —  3  ? 
First,  by  completing  the  square,  we  have, 

a^  4.  4a;  4-4  =   -3  +  4  =  1; 
and  by  extracting  the  square  root, 

a;  -f  2  =    ±  -/l   =   +1,   and    —  1 ; 
hence,  a;'=— 2  +  l  =  -l;  and  a;"=— 2-l  =  -3 
That  is,  in  this  form  both  the  roots  are  negative. 

VERIPICATION. 

If  we  take  the  first  negative  value,  the  equation, 
a;2  4.  4a;  =   —  8, 
gives  (-  1)24-  4(-  1)  =  1  -  4  =   -  8; 

and  by  talong  the  second  value,  the  equation, 
a;'  -f  4a;  =   ~  3, 


222  ELEMENTARY      ALGEBRA. 

gives  (—  3)2  +4(  -  3)   =  9  -  12  =  ^  3  ; 

hence,  both  values  of  x  satisfy  the  given  equation. 

2.  Wliat  are  the  vahies  of  aj  in  the  equation, 

-  |-  -  5a;  -  16  =  12  +  ia;2  ^  6a5? 

By  transposing  and  reducing,  we  have, 
—  a;2  —  lla;  =  28; 
then,  dividing  by  —  1,  the  coefficient  of  x\  vre  have, 

a;2  +  lla;  =   —  28; 
then,  by  completing  the  square, 

a;2  -h  Ux  -\-  30.25   =   2.25; 


hence,       a;  +  5.5   =    ±  -/2.25   =    +  1.5,    and    —  1.5 ; 
consequently,       a'  =   —  4,   and   a"  z=   —  7. 

3.  Wliat  are  the  values  of  x  in  the  equation, 

x^  1 

—  -  —  2x  —  5   =  -x'^  +  5x  +  5? 
8  8 

.        j  a;'    =    -  2. 
Ans.  ^ 


''\t' 


~  6. 

4.  What  are  the  values  of  a;  in  the  equation. 


2a;2+  8a;  =    -  2|  -  ?a;? 


.         (a;'    =    ~i 
(  a;"  =   --4 


5.  What  are  the  values  of  a;  in  the  equation, 

-  4a;2 


4a;2  -f-  ?aj  +  3a;  =   -  14a;  -  3|  -  4a;2? 


Ans.  J--     -        ^- 


COMPLETE      EQUATIONS.  233 

6.  What  are  the  values  of  as  in  the  equation, 
4  2 


^-iJ.iiJ; 


1.  What  are  the  values  of  jb  in  the  equation, 
^  +  7a;  -h  20  =   -  ?z2  -  11a;  -  60  ? 

9  a 


^'"•1J'=   ' 

-  8. 

-  10. 

8. 

What  are 

the  values  of  a: 

1  in  the 

equation. 

*-■ 

-.!  =  - 

9ia:- 

6           2 

-  8. 

9.  What  are  the  values  of  x  in  the  equation, 
lar»-f6a;+i=~^2-  5^x  -  ?  ? 

5  4  O  4 

^"^^  (  a;"  =   -  10. 

10.  What  are  the  values  of  a  in  the  equation, 

11.  What  are  the  values  of  a;  in  the  equation, 

aJ  +  4a;  —  90  =   —  93  ? 

A  J*'     =     -1. 

BTAMPLES    OF    TOE    POITRTH    FORM. 

I.  What  are  the  values  of  x  in  the  equation, 
a;2  -  8a;  =   -  7  ? 


224  ELEMENTARY       ALGEBRA. 

By  completing  the  square,  we  have, 

a;2  -  8a;  +  16   =   -  7  -f  16  =  ,9 
then,  by  extracting  the  square  root, 

a;  —  4  =   ±  y^  =   +  3,    and    —  3 ; 
hence,  x-  =   4-7,   and   aj"  =   +1. 

That  ig,  in  this  form,  both  the  roots  are  positive. 

VERIFICATION. 

It*  we  take  the  greater  root,  the  equation, 
a;2  _  8a5  =   —  7,     gives,    7^  —  8  X  7  =  49  —  56  :=  —  7; 
and  for  the  lesser,  the  equation, 

x^  —  8a;  =     —  7,     gives,    1^  —  8x1   =  1  —  8=    -7 
hence,  both  of  the  roots  will  satisfy  the  equation. 

2.  What  are  the  values  of  x  in  the  equation, 

—  \\x^  4-  3a;  -  10  =  \\x^  -  18a;  +  17? 

By  clearing  of  fractions,  we  have, 

—  3a;2  4-  6a;  —  20   =   2>x^  —  36a;  4-  40 ; 

then,  by  collecting  the  similar  terms, 

—  ^x^  4-  42a;  =  60 ; 

flien,  by  dividing  by  the  coefficient  of  a;'^,   which  is  —  6, 
we  have, 

a;2  -  7a5  =   -  10. 

By  completing  the  square,  we  have, 

a;^  —  7a;   f  12.25   —  2.25, 


COMPLETE      EQUATIONS.  225 

and  by  extracting  the  square  root  of  both  members, 

a;  -  3.5  =   ±  y^^  =   +  1.5,   and    —  1.5; 
hence, 

x'  —  3.5  4-  1.5   =  5,    and     aj"  =  3.5  —  1.5    =    2. 

VERIFICATION. 

If  we  take  the  greater  root,  the  eqnation. 
SB*  -  7x  =  -  10,    gives,    52  -  7  X  6  =  25  -  35  =  -  10: 
and  if  we  take  the  lesser  root,  the  equation, 
k2  —  7a;  =  —  10,      gives,     2^  —  7  X  2  =  4  —  14  -   — 10. 

3.  What  are  the  values  of  x  in  the  equation, 

-^  3a;  4-  2a;2  +  1   =   l7|a;  -  2a;2  -  3  ? 

By  transposing  and  collecting  the  terms,  we  have, 

.  4a;2  —  20|a;  =    —  4  ; 

then  dividing  by  the  coefficient  of  ai^,  we  have, 

a;2  -  6}a;  =   -  1. 

By  completing  the  square,  wc  obtain, 

,        ^,  169  ,    ,    169         144 

^-^i^+^   =    -^  +  -25-    =^5 

and  by  extracting  the  root, 
hence, 


,       «,  .      /r44  12  ,         12 


a:'  =  2|  +  -^   =  5,    and,    a;"  =  2J  -  \^   =  1. 
a  5  5 

VERIFICATION. 

If  we  take  the  greater  root,  the  eqnation, 
i'  —  5|a:  =  —  1,    gives,    o*  —  5|  x  5  =  26  —  26  —  —  1  ; 


226  ELEMENTARY      ALGEBRA. 

and  if  we  take  tLe  lesser  root,  the  equation, 
^-H^=-^    gives,  (^j  -  51  X  -=---=:-  1 
4.  What  are  the  values  of  x  in  the  equation, 


\^-^-  +  l  =  -f'  +  l--l-' 


Ans. 

5.  WTiat  are  the  values  of  a;  in  the  equation, 
1 

1 


ja'    =  3 


—  4a;2  _    a;  4-  i|   =    _  5x^  +  Sx? 


^^^•1^-=  t 


6.   WTiat  are  the  values  of  x  in  the  equation, 

_  4a;2  _}.  l_x =   —  3a;2  —  --a;  +  —  ? 

^  20  40  20     ^  40 


^^^•{J'=  t 


7.  What  are  the  values  of  x  in  the  equation, 


a;2  _  lOJ^JB    =    —  1  ? 


.        ja;'    =^   10. 


8.  What  are  the  values  of  x  in  the  equation, 

1 7'r2  97*2 

-  27a;  H    ^  +  100  =   ^  4-  12a;  -  26? 
5  5 


Ans 


j  a;'    =7. 
*  (  a"  -^-  0. 


9.  What  are  the  values  of  x  in  the  equation, 

ja;'    =  0. 

(  a:"    r=     I. 


22a;  +  15   = '—  +  28a!  -  30 

3  3 


Ans. 


PR0PEBTIK8     OF     EQUATIONS. 

10.  Wliat  are  the  values  of  as  in  the  equation, 

2aj»-30a;-f3  =   -  x' +  3tV«  -  ^^ 


Arts. 


\^    =--   11, 

(  X"  =   A. 


PROPERTIES    OF  EQUATIONS  OF  THE   SECOND    DEOREK 
FIRST  PROPERTY. 

161.  We  have  seen  (Art.  153),  that  every  complete 
equation  of  the  second  degree  may  be  reduced  to  the  form, 

a^  -I-  2px  =  q (1.) 

Completing  the  square,  we  have, 

transposing  q  +  p^  lo  the  first  member, 

x2  +  2pa;  +  /)2  -  (^^  -h  p")  =  0.        .     (2.) 
Now,  since  x^  +  2/>a;  +  p^  \&  the  square  of  sc  4-  7>,  and 
q  ^  p^  the  square  of  \/q~+  jt>*,  we  may  regard  the  first 
member  as  the  difference  between  two  squares.     Factoring, 
(Art.  66),  we  have, 

(aj  +  />  -h  -/  ?  4-  p')  {x  +  p  -  V7+~P)   =  0.     .     (  3.) 

Tliis  equation  can  be  satisfied  only  in  two  ways : 

1st.  By  attributing  such  a  value  to  a;  as  shall  render  the 
first  factor  equal  to  0  ;  or, 

Ifll.  To  what  form  may  every  equation  of  the  second  degree  be  re- 
duced? What  form  will  this  equation  take  after  completing  the  square 
and  transposing  to  the  first  member?  After  factoring?  In  bow  many 
ways  may  Equation  (  8  )  be  satisfied  ?  What  are  they  ?  How  many  roots 
has  every  equation  of  tho  «orftT ''  -i-^-,  ■>  ? 


228  ELEMENTARY       ALGEBRA. 

2d.  By  attributing  such  a  value  to  x  as  shall  render  the 
Becond  factor  equal  to  0. 

Placing  the  second  factor  equal  to  0,  we  have, 

aj  ^-p  —  ^/q  ■\-i9-  —  0;   and  a'  =  —  p -\-  Vq~+^*        (4.) 

Placing  the  first  factor  equal  to  0,  we  have, 


■i-p  1'  yq+2^^  =  ^'i   ^^^   a;"  =  —p  —  y/q  ^p\       (6,) 

Since  every  supposition  that  will  satisfy  Equation  (  3  ),  will 
also  satisfy  Equation  ( 1 ),  from  which  it  was  derived,  it  fol- 
lows, that  x'  and  a"  are  roots  of  Equation  ( 1  ) ;  also,  that 

Every  equation  of  the  second  degree  has  two  roots,  and 
only  two, 

Note. — ^The  two  roots  denoted  by  a;'  and  a",  are  the 
same  as  found  in  Art.  158. 

SECOND    PROPERTY. 

162.  We  have  seen  (Art.  161),  that  every  equation  of 
the  second  degree  may  be  placed  under  the  form, 

{x  -\-  p  +  -y/q  +  p^)  {x  -^  p  —  Vq~^^^)   =  0- 

By  examining  this  equation,  we  see  that  the  first  factor 
may  be  obtained  by  subtracting  the  second  root  from  the 
unknown  quantity  a; ;  and  the  second  factor  by  subtracting 
the  j^rs^  root  from  the  unknowoi  quantity  jb;  hence, 

JEoery  equation  of  the  second  degree  may  he  resolved  into 
two  binomial  factors  of  the  first  degree,  the  first  terms,  in 
both  factors,  being  the  unknoicn  quantity,  and  the  second 
terms,  the  roots  of  the  equation,  taken  loith  contrary  sigyis. 

162.  Into  how  many  binoniinl  factors  of  the  first  degree  may  every 
equation  of  the  second  degree  be  resolved?  What  are  the  first  terms  of 
thcee  factors  ?     What  the  second  ? 


FOKMATION       OF      EQUATIONS.  229 


TIIIUD     PROPERTY. 


163.     Il' we  add  Equations  (4)  and  (5),  Art.  161,  we 
have,  

a'    =    —  p  4    ^/q  -j-  p^ 


." 


=  -  />  -  V?  +  i>^ 


as'  4-  a:"  =   —  2/> ;  that  is. 

In  every  reduced  equation  of  (he  second  degree^  the  sum 
of  the  two  repots  is  equal  to  the  coefficient  of  the  second  term  \a^/^ 
taken  with  a  contrary  sign. 


FOURTH     PPvOPERTY. 


164.     If  we  multiply  Equations  (4)  and  (5),  Art.  161, 
member  by  member,  we  have. 


x'  X  x"  =  (—  7>  4-  Vq  4-  p^)  {-  P  —  V5-KP^) 
=  jo2  -  (2-  +  jt)2)  =   _  5^;    that  is, 

In  every  equation  of  the  second  degree^  the  product  of 
the  two  roots  is  equal  to  the  knoimi  term  in  the  second  mem- 
ber^ taken  with  a  coiitrary  sign. 


FORMATION  OF  EQUATIONS  OF  THE  SECOND  DEGREE. 

165.  By  taking  the  converse  of  the  second  property, 
i \y\.  102),  we  can  form  equations  which  sliall  have  given 

iliat  is,  if  they  are  known,  \\  <•  cm   liiid  the  corre- 
gpouding  equations  by  the  following 

RULE.  ^ 

L   Subtract  each  root  from  the  tmknown  quantity :         ^ 

163.  What  is  the  algebraic  ecm  of  the  roots  equal  to  in  every  eqiiatioii 
of  the  second  degree  ? 

164.  What  is  the  product  of  the  roots  equal  to? 

166.  Ifow  will  yon  6nd  th#«!qiifttion  whor  the  roots  ar€  known  ? 


230  ELEMENTARY       ALGEBRA. 

n.   Multiply  the  results  together^  and  place  their  prodiict 
equal  to  0. 


^, 


EXAMPLES. 


Note. — Let  the  pupil  prove,  in  every  case,  that  the  roots 
will  satisfy  the  third  and  fourth  properties. 


1.  If  the  roots  of  an  equation  are  4  and  —  5,  what  is  the 
equation  ?  Ans.   x^  -{-  x  =  20. 

2.  What  is  the  equation  when  the  roots  are  1  and  —  3  ? 

A71S.   cc2  -f  2a;  =  3. 

3.  What  is  the  equation  when  the  roots  are  9  and  —  10  ? 

A?is.   x'^  -\-  X  =  90. 

4.  What  is  the  equation  whose  roots  are  6  and  —  10? 

Ans.   x^  -\-  4:X  =  60. 

6.  Wliat  is  the  equation  whose  roots  are  4  and  —  3  ? 

Ans.   x"^  —  X  =z  12. 

6.  What  is  the  equation  whose  roots  are  10  and  —  y^  ? 

Ans.   x^  —  9-i-\£c  =   1. 

7.  What  is  the  equation  whose  roots  are  8  and  —  2  ? 

Ans.   a;2  —  6a;  =:  16. 

8.  What  is  the  equation  whose  roots  are  16  and  —  5  ? 

Ans.   x^  —  11a;  z=   80 

9.  What  is  the  equation  whose  roots  are  —  4  and  —  5 ! 

Ans.   a;2  4-  9a;  =   —  20. 

to.  What  is  the  equation  whose  roots  are  —  6  and  —  7  ? 

Ans.   a;2  +  13a;  =   —  42. 

g 

11.  What  is  the  equation  wliose  roots  are  —  -  and  --  2  ? 

q 

Ans.  a;2  -f  2|a;  =   —  -. 

12.  What  is  the  equation  whcse  roots  are  —  2  and  —  3  ? 

Ans.   a;^  4-  5a;  =    —  6. 


NUMERICAL     VALUES     OF     THE     ROOTS.       231 

18.  What  is  the  equation  whose  roots  are  4  and  3  ? 

Ans.  x^  —  1x  z=  —  12. 

14.  What  is  th^  equation  whose  roots  are  12  and  2  ? 

Afis.   x^  -  \4x  =   —  24, 

16.  What  is  the  equation  whose  roots  are  18  and  2? 

A^^s.    x^  —  20a;  =    —  30. 

16.  Wliat  jp  the  equation  whose  roots  are  14  and  3? 

Ans.   a;2  —  1 7a;  =  .  —  42. 

4  9 

1 1,  What  is  the  equation  whose  roots  are  -  and  —  t  ? 

An^:   x^  H X  =   1, 

'  2 

18.  What  is  the  equation  whose  roots  are  5  and  —  „  ? 

Ans.   x^ -X  —   -    • 

3  u 

19.  What  is  the  equation  whose  roots  are  n  ^  nnd  h  ? 

Ans.   a;2  —  (a  -f  b)x  =  —  ab. 

20.  What  is  the  equation  whose  roots  are  c  and  —  d? 

Ans.  a;*  —  (c  —  d)x  =  c€L 


TRINOMIAL     EQUATIONS     OF    THE    SECOND     DEGREE. 

165.*  A  trinomial  equation  of  the  second  degree  con- 
tains three  kinds  of  terms : 

1st.  A  term  involving  the  unknown  quantity  to  the  second 
degree. 

2d.  A  term  mvolving  the  unknown  quantity  to  the  first 
degree ;  and 

3d.  A  known  term.     Thus, 

a;2  -  4a:  -  12   =  0, 
10  a  trinomial  equation  cf  the  second  degree. 


232  ELEMENTARY       ALGEBRA. 

FACTOEING, 

165.**    What  are  the  factors  of  the  trinomial  equation, 

a;2—  4a;  —  12  =  0? 

A  trinomial  equation  of  the  second  degree  may  always  be 
reduced  to  one  of  the  four  forms  (Art.  160),  by  simply  trans- 
posing the  known  term  to  the  second  member,  and  then 
solving  the  equation.  Thus,  from  the  above  equation,  we 
have, 

x^  —  Ax  =z   12. 

Resolving  the  equation,  we  find  the  two  roots  to  be  +6 
and  —  2 ;  therefore,  the  factors  are,  x  —  6,  and  a;  4-  2 
(Art.  162). 

Since  the  sum  of  the  two  roots  is  equal  to  the  coefficient 
of  the  second  term,  taken  with  a  contrary  sign  (Art.  163) ; 
and  the  product  of  the  two  roots  is  equal  to  the  known 
term  in  the  second  member,  taken  with  a  contrary  sign,  or 
to  the  third  term  of  the  trinomial,  taken  with  the  same 
sign :  hence  it  follows,  that  any  trinomial  may  be  factored 
by  inspection,  when  two  numbers  can  be  discovered  whose 
algebraic  sum  is  equal  to  the  coefficient  of  the  second  term^ 
and  whose  product  is  equal  to  the  third  term, 

EXAMPLES 

1.  Wbat  are  the  factors  of  the  trinomial,  cc^  _  g^.  _  3gp 

It  is  seen,  by  inspection,  that  —  1 2  and  +  8  will  fulfil  the 
conditions  of  roots.  For,  12  —  3  —  9;  that  is,  the  co 
efiicient  of  the  second  term  with  a  contrary  sign ;  and 
12  X  —  3  =  —  36,  the  third  term  of  the  trinomial;  hence, 
the  factors  are,  jb  —  12,  and  a;  +  3. 

2.  What  are  the  factors  of  a:^  _  7^;  —  30  =  0  ? 

Ans.   ic  —  1 0,   and   ar  + 


TRINOMIAL      EQUATIONS  233 

8.  What  are  the  factors  of  x^  4-1535+36  ==  0?  , 

Ans.   X  +  12,  and  a;  -h  3. 

4.  What  are  the  factors  of  a^  —  12a;  —  28  =  0  ? 

Ans,  X  —  11,   and  a;  -f-  2 

5.  What  are  the  factors  of  a;^  _  ^a;  —  6  =  0  ? 

Ana,  a;  —  8,   and  a;  +  1. 


TRINOMIAL  EQUATIONS   OF  THE    FOEM 

x^*  4-  Ipx""  —  q. 

In  the  above  equation,  the  exponent  of  a,  in  the  first  term, 
is  double  the  exponent  of  aj  in  the  second  term.* 

a;fi  —  4a;3  =  32,    and    a*  +  4a;2  =  117, 

are  both  equations  of  this  form,  and  may  be  solved  by  the 
rules  already  given  for  the  solution  of  equations  of  the 
second  degree. 
In  the  equation, 

aj2«  -f  2/xc"  =  <?, 

we  see  that  the  first  member  will  become  a  perfect  square, 
by  adding  to  it  the  square  of  half  the  coefficient  of  a;" ;  thus, 

a;2n  _|.  2jt?a;'»  +  p^  =  q  +  p\ 

in  which  the  first  member  is  a  perfect  square.    Then,  ex- 
tracting the  square  root  of  both  members,  we  have, 


«•  +  p  =  ±  Vq+^ ; 


hence,  a:»  =  —  p  ±  Vq-\-p^\ 

then,  l)y  taking  the  nth  root  of  both  members, 


x'   =  V-Jt)4-  V^~+>S 


and  Jc"  =  V-io  -  V-  P  ^  P" 


234  ELEMENTARY       ALGEBKA. 

•  EX  A  MPLES. 

1.  What  are  the  values  of  a;  in  the  equation. 

a;6  -f  Qx3  =  112? 
Completing  the  square, 

aj6  +  6053  +  9  =  112  +  9  =  12I  ; 
then,  extracting  the  square  root  of  both  members, 

053  +  3  =   ±  -/m  ~   ±  11 ;  hence, 
a'  =  3y_  3  ^  11^     and    a"  =  ^—  3  —  11 ;  hence, 
35'  =  3^8  =  2,  and    a"  =  y-  14  =   -  y/lA. 

2.  What  are  the  values  of  05  in  the  equation, 

X"  -  Sx^  =  9  ? 
Completing  the  square,  we  have, 

05*  —  8052  +  16  =  9  +  16   =  25. 
Extracting  the  square  root  of  both  members, 
a;2  _  4  _    -j-  y/25   =   ±  5  ;  hence. 


05'  =    ±  V4  +  5,     and     05"  =    =h  -v/4  —  5  ;  hence. 


a;'  =  +  3  and  —  3  ;    and    05''  =  +  y  —  1  and  —  y^  1. 
3.  What  are  the  values  of  0;  in  the  equation, 

0J6  +  20;k^  =   69  ? 
Completing  the  square, 

056  +  20o;3  +100   =   69  +  100  =   169. 
Extracting  the  square  root  of  both  members, 

053  +  10  =    ±  -/leg  =    ±  13  ;  hence, 
x"  =  \J-  10  -f  18,     and    05"  =  Ij-  10  ~  13. 
05'  =  \fz,     and     05"  —  ^''—  23. 


TRINOMIAL      EQUATIONS.  235 

4.  What  are  the  values  of  a;  in  the  equation, 
»♦  —  2ib2  =  3  ? 
Ans,  x"  z=   ±  y^,   and  a/'  =  ±  y/^, 

6.  What  are  the  vahies  of  2  in  the  equation, 


Ans.   x'  =  1,   and   x"  =  y--^. 


6.  Given  x  ±  yOa  4-  4   =   12,   to  find  x. 
Transposing  x  to  the  second  member,  and  then  squaring, 
9a;  -h  4  =  a;2-  24a;  +  144; 
.-.     a;2  —  33a;  =   —  140; 
and,  x'  =  28,     and    x"  —  5. 


7.   4a;  ±  Ay/x  -f  2  =  7.  ^/w.   aj'  =  4^,   x"  =  i. 


8.   a;  ±  -/5a;  -|-  10  =  8.  Ans.   x*  =  18,   a;"  =  3. 


NUMERICAL     VALUES      OF     TIIE     ROOTS. 

166.     We  have  seen  (Art.  IGO),  that  by  attributing  all 

possible  signs  to   2/>  and  ^,    ^^t'  liave  the  four  following 

forms: 

a;2+  2;xB  =  ^ (1.) 

a;2  —  'Ipx  —  q (2.) 

a;2_,.  27a  =    -  ^ (3.) 

7?  —  2jyx  =    —  q (4.) 

166.  To  how  many  forms  may  every  equation  of  the  second  degree  be 
reduced  ?    What  are  they  ? 


236  ELEMENTAR-S       ALGEBRA. 


First  Form, 

167.  Since  q  is  positive,  we  know,  from  Property 
Fourth,  that  the  product  of  the  roots  must  be  negative  • 
hence,  the  roots  have  contrary  signs.  Since  the  coefficient 
2p  is  positive,  we  know,  from  Property  Third,  that  the  alge- 
braic sum  of  the  roots  is  negative  ;  hence,  the  negative  root 
is  numerically  the  greater. 

Second  Form. 

168.  Since  q  is  positive,  the  product  of  the  roots  must 
be  negative;  hence,  the  roots  have  contrary  signs.  Since 
2/?  is  negative,  the  algebraic  sum  of  the  roots  must  be  posi* 
tive  ;  hence,  the  positive  root  is  numerically  the  greater. 

Third  Form. 

169. .  Since  q  is  negative,  the  product  of  the  roots  is 
positive  (Property  Fourth) ;  hence,  the  roots  have  the  same 
sign.  Smce  2p  is  positive,  the  sum  of  the  rooih  must  be 
negative ;  hence,  both  are  7iegative. 

Fourth  Form, 

170.  Since  q  is  negative,  the  product  of  the  roots  is 
positive ;  hence,  the  roots  have  the  same  sign.  Since  2p  is 
negative,  the  sum  of  the  roots  is  positive ;  hence,  the  roots 
are  both  positive. 

167.  What  sign  has  the  product  of  the  roots  in  the  first  form?  IIow 
are  their  signs?     Which  root  is  numerically  the  greater?     Why? 

168.  What  sign  has  the  product  of  the  roots  in  the  second  form  f  How 
are  the  signs  of  the  roots  ?     Which  root  is  numerically  the  greater  ? 

169.  What  sign  has  the  product  of  the  roots  in  the  third  form?  How 
are  their  signs  ? 

170.  What  sign  has  the  product  of  the  roots  in  the  fourth  form  ?  How 
are  the  signs  of  the  roots  ? 


NUMEEICAL     VALUE     OF     THE     BOOTS.         237 

I^irst  and  Second  Forms, 

ITl,     If  we  make  $^  =  0,  the  first  form  becomes, 

a;2  4-  2/XB  =  0,     or    x{qc  +2/))  =  0 ; 

ttrhich  shows  that  one  root  is  equal  to  0,  and  the  other  to  —2/1 

Under  the  same  supposition,  the  second  form  becomes, 

7?  —  Ipx  =  0,    or    x{x  —  2/?)  =  0 ; 

which  shows  that  one  root  is  equal  to  0,  and  the  other  to 
2p.  Both  of  these  results  are  as  they  should  be ;  since,  when 
7,  the  product  of  the  roots,  becomes  0,  one  of  the  factors 
must  be  0 ;  and  hence,  one  root  must  be  0. 

Third  and  Fourth  Forms, 

172.  If,  in  the  Third  and  Fourth  Forms,  q^p^^  the 
quantity  under  the  radical  sign  will  become  negative  ;  hence, 
its  square  root  cannot  he  extracted  (Art.  13 7).  Under  this 
supposition,  the  values  of  x  are  imaginary.  How  are  these 
results  to  be  interpreted  ? 

If  a  given  number  he  divider  into  tico  imrts^  their  pro- 
duct will  he  the  greatest  possible^  when  the  parts  are  equal. 

Denote  the  number  by  2p,  and  the  difference  of  the  parts 
by  d\  then, 

p  -\-  -  =.    the  greater  part,     (Page  120.) 
and,  jt)  —  -   =    the  less  part, 

and,  p^  —  —  =    P,  their  product. 


171.  If  we  make  y  =  0,  to  what  does  the  first  form  reduce?  What, 
then,  are  ita  roots  ?  Under  the  same  supposition,  to  what  does  the  second 
form  reduce  ?    What  are,  then,  its  roots  ? 

1*72.  If  ^  >  p\  in  the  third  and  fourth  forms,  what  takes  place? 

If  a  number  be  divided  into  two  parts,  when  will  the  product  be  the 
greatest  possible  ? 


238  ELEMENTARY       ALGEBRA 

It  is  plain,  that  the  product  P  will  increase^  as  d  dimirir 
ishes,  and  that  it  will  be  the  greatest  possible  when  d  =  0; 
for  then-  there  will  be  no  negative  quantity  to  be  subtracted 
from  jo^,  in  the  first  member  of  the  equation.  But  when 
d  =  Oj  the  parts  are  equal ;  hence,  the  product  of  the  two 
parts  is  the  greatest  when  they  are  equal. 

In  the  equations, 

a;2  +  2pa;  =   —  <^,       a^  —  2j!%c  =   —  g', 

2/>  is  the  sum  of  the  roots,  and  —  q  their  product ;  and 
hence,  by  the  principle  just  estabhshed,  the  product  q^ 
can  never  be  greater  than  p^.  This  condition  fixes  a  limit 
to  the  value  of  q.  If,  then,  we  make  q  >  j^^^  ^ye  pass  this 
limit,  and  express,  by  the  equation,  a  condition  which  cannot 
be  fulfilled  ;  and  this  incompatibiUty  of  the  conditions  is 
made  apparent  by  the  values  of  x  becoming  imaginary. 
Hence,  we  conclude  that, 

"When  the  values  of  the  unknown  quantity  are  imaginary ^ 
the  conditions  of  the  proposition  are  incompatible  with 
each  other, 

EXAMPLES. 

1.  Find  two  numbers,  whose  sum  shaU  be  12  and  pro- 
duct 46. 

Let  X  and  y  be  the  numbers. 

By  the  1st  condition,  x  -\-  y  =  \1\ 

and  by  the  2d,  xy  =  46. 

The  first  equation  gives, 

X  =  \2  —  y. 

Substituting  this  value  for  x  in  the  second,  we  have. 

12y  -2/2  =  46; 

and  changing  the  signs  of  the  terms,  we  have, 

y^  -  \1y   =    —  46 


NUMEBICAL     VALUE     OF     THE     KOOT6.         239 

Then,  by  completing  the  square, 

y2  -  12y  +  36  =   -  46  +  36  =   -  lOj 
which  gives,  y'    =  ^  +  V^—  10, 


and.  y"  =  6  —  -/-  10; 

!)Oth  of  which  values  are  imaginary,  as  indeed  they  shoold 
l>e,  since  the  conditious  are  incompatible. 

2.  The  sum  of  two  numbers  is  8,  and  their  product  20 : 
what  are  the  numbers  ? 

Denote  the  numbers  by  x  and  y. 
By  the  first  condition, 

35  +  2/  =  8; 
and  by  the  second,  sey  =  20. 

The  first  equation  gives, 

a;  =  8  —  y. 
Substituting  this  value  of  as  in  the  second,  we  have, 

Sy  -y^  =  20 ; 
changing  the  signs,  and  completing  the  square,  we  have, 

y2-  8y  -f  16  =   -4; 
and  by  extracting  the  root, 

y'  =  4  -t-  \/^^,    and    y"  =  4  -  y/"^^. 
These  values  of  y  may  be  put  under  the  forms  (Art.  142), 
y  =  4  +  2-/^,    and    y  =   4  —  2>/^^. 

3    What  are  the  values  of  sc  in  the  equation, 
«2+  2a;  =   -  10? 


ix"  =    -  1 


240  ELEMENTAKY      ALGEBKA. 


peoble:ms. 

1.  Find  a  number  such,  that  twice  its  square,  added  to 
three  times  the  number,  shall  give  65. 

Let  X  denote  the  unknown  number.    Then,  the  equation 
of  the  problem  is, 

2-052  4-  3a;  =:  65  ; 
whence, 

3    ,       /G5    ,     9  3    ,    23 

^=  -i^Vy  +  Ie  =  -4±T- 

Therefore, 

3        23  ^  ,        ,,  3        23  13 

x^  =--  +  -  =  5,    and    x-=   ----=  --. 

Both  these  values  satisfy  the  equation  of  the  problem. 
For, 

2  X  (5)2  4-3x5=:2x25  4-15   =  65; 

:i       J      13\2  ,    ^  13         169        39         130 

^d,      2(-  -)  +  3  X  -  -   =  —  -  -  =  -   =  65. 

Notes. — 1.  If  we  restrict  the  enunciation  of  the  problem 
to  its  arithmetical  sense,  in  which  "  added  "  means  numer- 
ical i7icrease,  the  first  value  of  x  only  will  satisfy  the  con- 
ditions of  the  problem. 

2.  If  we  give  to  "  added,"  its  algebraical  signification 
(when  it  may  mean  subtraction  as  well  as  addition),  the 
problem  may  be  thus  stated : 

To  find  a  number  such,  that  twice  its  square  diminished 
by  three  times  the  number,  shall  give  65. 

Tlie  second  value  of  x  will  satisfy  this  enunciation ;  for, 


m 


13         169        39 


PBOBLBMS.  24] 

8.  The  root  which  results  from  giving  the  plus  sign  to  the 
mdical,  is,  generally,  an  answer  to  the  question  in  its  arith- 
metical sense.  The  second  root  generally  satisfies  the  pro- 
blem under  a  modified  statement. 

Thus,  in  the  example,  it  was  required  to  find  a  number, 
of  which  twice  the  square,  added  to  three  times  the  num- 
ber, shall  give  65.  Now,  in  the  arithmetical  sense,  added 
means  increased ;  but  in  the  algebraic  sense,  it  implies  dimi- 
nution when  the  quantity  added  is  negative.  In  this  sense, 
the  second  root  satisfies  the  enunciation. 

2.  A  certain  person  purchased*  a  number  of  yards  of  cloth 
for  240  cents.  If  he  had  purchased  3  yards  less  of  the  same 
cloth  for  the  same  sum,  it  would  have  cost  hhn  4  cents  more 
per  yard :  how  many  yards  did  he  buy  ? 

Let  X    denote  the  number  of  yards  purchased. 

240 
Then,   —    will  denote  the  price  per  yard. 

H  for  240  cents,  he  had  purchased  three  yards  less,  that 

v\.  a;  —  3  yards,  the  price  per  yard,  under  this  hypothesis, 

240 
would  have  been  denoted  by •    But,  by  the  condi- 

tioiiK,  tliis  last  cost  must  exceed  the  first  by  4  cents.     There- 
fore, we  have  the  equation, 

240  240   _ 

JB- 3         a;     -  *' 
whence,  by  reducing;       7^  —  2x  =  180, 


and,  x=  -±^-  +  180  =  —^i 

therefore,  jc'  =   15,    and    jb"  =   —  12. 

Notes. — 1.  Tlie  value,  a;'  =  15,  satisfies  the  enunciation 
in  its  arithmetical  sense.     For,  if  16  yards  cost  240  cents, 
11 


2  42  ELEMENTARY       ALGET-  RA. 

240  -7-15  =  16  cents,  the  price  of  1  yard ;  and  240  --  12  =  20 
cents,  the  price  of  1  yard  under  the  second  supposition. 

2.  The  second  value  of  x  is  an  answer  to  the  following 
Problem : 

A  certain  person  purchased  a  number  of  yards  of  cloth 
for  240  cents.  If  he  had  paid  the  same  for  three  yards  more, 
it  would  have  cost  him  4  cents  less  per  yard :  how  many 
yards  did  he  buy  ? 

This  would  give  the  equation  of  condition, 
240  240 

x^  —  Sx  =  180; 

the  same  equation  as  found  before  ;  hence, 

A  single  equation  will  often  state  tioo  or  more  arith- 
metical problems. 

T'liis  arises  from  the  fact  that  the  language  of  iVlgebra  is 
more  comprehensive  than  that  of  Arithmetic. 

3.  A  man  having  bought  a  horse,  sold  it  for  $24.  At  the 
sale  he  lost  as  much  per  cent,  on  the  price  of  the  horse,  as 
the  horse  cost  him  dollars :  what  did  he  pay  for  the  horse  ? 

Let  X  denote  the  number  of  dollars  that  he  paid  for  the 
horse.     Then,  a;  —  24  will  denote  the  loss  he  sustained.    But 

X 

M  he  lost  X  per  cent,  by  the  sale,  he  must  have  lost  -— - 

opon  each  dollar,  and  upon  a;  dollars  he  lost  a  sura  denoted 
3.2 

by    ;   we  have,  then,  the  equation, 

•^     100  7  J  » 

—  =  SB  -  24;    whence,    x^  -  100a;  =    -  2400, 


rROBLEMS.  243 

and,  X  =:  50  ±  y^OO  —  2400  =  50  ±  10. 

nierefore,  7f  =  CO,     and    a"  =40. 

Botli  of  these  roots  will  satisfy  the  problem. 

For,  if  the  man  gave  $60  for  the  horse,  and  sold  him  for 
|24,  he  lost  $36.  From  the  enunciation,  he  should  have  lost 
GO  per  cent,  of  $60  ;  that  is, 

60      .  ^^         60  X  60 

—  of  60  = =  36  ; 

100  100  ' 

therefore,  $60  satisfies  the  enunciation. 

Had  he  paid  $40  for  the  horse,  he  would  have  lost  by  the 
sale,  $16.  From  the  enunciation,  he  should  have  lost  40  pei 
cent,  of  $40 ;  that  is, 

40      ^  ,^         40  X  40         ,^ 

—  of  40  = =  16  : 

100  100 

therefore,  $40  satisfies  the  enunciation. 

4.  The  sum  of  two  numbers  is  11,  and  the  sum  of  their 
squares  is  61 :  what  are  the  numbers?  Ans,   5  and  6 

5.  The  difference  of  two  numbers  is  3,  and  the  sum  of  their 
squares  is  89  :  what  are  the  numbers  ?  Ans.   5  and  8. 

6.  A  grazier  bought  as  many  sheep  as  cost  him  £60,  and 
after  reserving  fifteen  out  of  the  number,  he  sold  the  re- 
mainder for  £54,  and  gained  25.  a  head  on  tliose  he  sold  : 
how  many  did  he  buy  ?  Ans.    75. 

7.  A  mercliant  bought  cloth,  for  which  he  paid  £33  15a., 
which  he  sold  again  at  £2  8«.  per  piece,  and  gained  by  tlie 
liargain  as  much'  as  one  piece  cost  him :  how  many  pieces 
did  he  buy?  Ans.    15. 

8.  The  difference  of  two  numbers  is  9,  and  their  sum, 
mnltiplied  by  the  greater,  is  equal  to  266:  what  are  the 
numbers?  Ans.    14  and  5 


244  ELEMENTARY      ALGEBKA. 

9.  To  find  a  number,  such  that  if  you  subtract  it  from  10^ 
and  multiply  the  remainder  by  the  number  itself,  the  pro- 
duct will  be  21.  A71S.   7  or  3. 

10.  A  person  traveled  105  miles.  If  he  had  traveled  2 
miles  an  hour  slower,  he  would  have  been  6  hours  longer  in 
completing  the  same  distance :  how  many  miles  did  he  travel 
per  hour  ?  A7is.    1  miles. 

11.  A  person  purchased  a  number  of  sheep,  for  which  he 
paid  $224.  Had  he  paid  for  each  twice  as  much,  plus  2  dol- 
lars, the  number  bought  would  have  been  denoted  by  twice 
what  was  paid  for  each  :  how  many  sheep  were  purchased  ? 

Ans.    32. 

12.  The  difference  of  two  numbers  is  7,  and  their  sum 
multiplied  by  the  greater,  is  equal  to  130  :  what  are  the 
numbers?  A?is.    10  and  3. 

13.  Divide  100  into  two  such  parts,  that  the  sum  of  their 
squares  shall  be  5392.  A7is.    64  and  36. 

14.  Two  square  courts  are  paved  with  stones  a  foot  square ; 
the  larger  court  is  12  feet  larger  than  the  smaller  one,  and 
the  number  of  stones  in  both  pavements  is  2120  :  how  long 
is  the  smaller  pavement  ?  Ans.    26  feet. 

15.  Two  hundred  and  forty  dollars  are  equally  distributed 
among  a  certain  number  of  persons.  The  same  sum  is  agam 
distributed  amongst  a  number  greater  by  4.  In  the  latter 
case  each  receives  10  dollars  less  than  in  the  former:  how 
many  persons  were  there  in  each  case.  A7is.    8  and  1 2. 

16.  Two  partners,  A  and  B,  gained  360  dollars.  A^s 
money  was  in  trade  12  months,  and  he  received,  for  prin- 
cipal and  profit,  520  dollars.  B^s  money  was  600  dollars, 
and  was  in  trade  16  months:  how  much  capital  had  A  ? 

Ans.    400  dollars. 


MOBS    THAN    ONB    UNKNOWN    QUANTITY.      245 


■QUATIONS  INVOLTrNG  M  :RE  THAN  ONE  UNKNOWN  QUANTITT. 

173.  Two  simultaneous  equations,  each  of  the  second 
degree,  and  containing  two  unknown  quantities,  will,  when 
combined,  generally  give  rise  to  an  equation  of  the  fourth 
degree.  Hence,  only  particular  cases  of  such  equations  can 
be  solved  by  the  methods  already  given. 

FIRST. 

7\co  simidtitneouR  eqtiations^  involviiig  two  unknown 
quantitiesy  can  always  he  solved  when  one  is  of  the  first 
and  tlie  other  of  tlie  second  degree, 

E  X  A  jr  P  L  E  s . 

1.  Given      \    .,       ^  ~ \     to  find  x  and  y. 

By  transposing  y  in  the  first  equation,  we  have, 
JB  =   14  -  y; 
and  by  squaring  both  members, 

xt  =   196  —  28y  +  y'\ 

Substituting  this  value  for  7?  in  the  second  equatioDy  we 
have, 

196  —  28y  +  y2  4.  y2  _   joq. 

from  wliich  we  have, 

y2  _  i4y  =   ~  48. 

By  completing  the  square, 

y2  —  14y  -f  49  =  1  ; 

178.  When  nuiy  two  simultaneous  equations  of  the  second  degree  be 
eoived  ? 


246  ELEMENTARY       ALGEBRA. 

and  by  extracting  the  square  root, 

y  -  7   =    ±  -/l   =    +  1,     and     -  1  ; 
hence,  y'  =   7  +  1   =   8,     and     y"  —  7  —  1   =  6. 

If  we  take  the  greater  vahie,  we  find  a;  =  6  ;  and  if  we 
take  the  lesser,  we  find  a;  =  8. 

{x'  =   8,    x"  =r.   6 

VERIFICATION. 


For  the  greater  value,  y  =  8,  the  equation, 

X    -\-  y    z=z     14,     gives       6  +     8  =  14; 

and,        a;2  +  2/2  _   iqq,     gives     36  +  64   —  100. 
For  the  value  y  —  6,  the  equation, 

X   -\-  y    =     14,.   gives       8  +    6  =  14; 

and,        x^  +  y"^  =   100,     gives     64  +  36   =  100. 

Hence,  both  sets  of  values  satisfy  the  given  equation. 

2.  Given   i    ^      '^^        . .  r  to  find  x  and  y. 
(  a;2  —  2/2  _  45  )  i^ 

Transposing  y  in  the  first  equation,  we  have, 
a;  =  3  +  y ;      • 
then,  squaring  both  members, 

x'  =:   9  +  6?/  +  y\ 

Substituting  this  value  for  £c^,  in  the  second  equation,  w^ 

have, 

9  +  62/  +  2/'- y^  -=   45; 

whence,  we  have, 

Qy  —   36,    and    y    =  6. 


8IMDLTANEOU8      EQUATIONS.  24:7 

SnbBtituting  this  value  of  y,  in  the  first  equation,  we  have, 
aj  —  6  =  8, 
and,  consequently,     aj'  =  3  +  6   =  9. 

VERIFICATION. 

X  —  ij    z=     3,    gives      9  —    6  =     3 ; 
and,  a;2—  2/2  _  45^    giygg    81  —  36  =  45. 

Solve  the  follo\ving  simultaneous  equations : 

]  ar^+  2/2  ^   117  f  (  y'  =  6,    y"=    -  9. 

ja;-f-y  =   9  I  ja;'=5,    ^"=   5. 

•  -1  x^-  2X7/  +  y2^   i\         ^^^-  I  y'  =  4,    y"=:  4. 

(flj-y  =  5  I 

•  (  jbM-  2a^  +  y2  ^  225  f 

ix'  =   10,    a;"=   —    5. 
I  y'  =     5,    y"=   -  10. 

SECOND. 

174,  7\ro  simtUtaneous  equations  of  the  second  degree^ 
which  are  homogeneous  with  respect  to  the  unknown  quari' 
tity^  can  always  be  solved. 

EXAMPLES* 

,    p.  J  Jc2_^  3a^  _  22  (1.) 

1.  l^iven   ]a.2  4.  3ay  +  2y2  =  40 (2.) 

to  find  X  and  y. 

174.  When  may  two  sunuiuueotis  eqrataons  of  the  second  degree  be 
solved  ? 


24S  ELEMENTARY      ALGEBRA. 

Assume  oi  —  ty^  t  being  any  auxiliary  unknown  quantity. 
Substituting  this  value  of  x  in  Equations  (  1 )  and  (  2  ), 
we  have, 

«y+3«y^=22,         ...     y2=_-_;  (3.) 

40 
ey  4    3^y^  +  2y2  ^   40,  .'.     y2  ^   ___^___ ,     (4) 

22  40 

nence, 


^2  +  3^  —   f^  +  Zt  +  2' 
hence,  22«2  +  QQt  +  44   =  40^^  _|_  120^; 

22 


reducing,  f^  ■{-  ^t  = 


2  11 

whence,  t'  =  - ,    and    t''= • 

'  3'  3 

Substituting  either  of  these  values  in  Equations  (  3  )  or 
(  4  ),  we  find, 

2/'  =   +3,    and    y"  =   —  3 

Substituting  the  plus  value  of  y,  in  Equation  (1),  ^e 
have, 

x^  -{-  9x  z=   22  ; 
fi'om  which  we  find, 

x'  =    +2,    and    cc"   =    —  11. 

If  we  take  the  negative  value,    y"  =^    —  3,   we  have, 
from  Equation   (  1 ), 

a;2  —  9ic  =  22  ; 
from  which  we  find, 

a;'   =    +  11,    and    a"   =   —  2. 

VERIFICATION. 

For  the  values    y'  =   +3,   and  x'  =   +2,   the  given 
equation, 

aj2  4.  3a;y  =  22, 


SIMULTANEOUS      EQUATIONS.  249 

gives,  22 +3x2x3   =  4  +  18  =  22; 

and  for  the  second  value,  a"  =   —  11,   the  wime  equation, 

a;2  +  Zxy  =  22, 
gives,     (-  11)2+  3  X  -  11  X  3   =  121  -  99  =  22. 

If,  now,  we  take  the  second  value  of  y,  that  is,  y"  =  —  3, 
and  the  corresponding  values  of  a^  viz.,  x'  =  +11,  and 
a:"  =   —  2;    for    x'  =   +11,    the  given  equation, 

a;2  +  3icy  =  22, 
gives,       112  +  3  X  11  X  —  3   =  121  —  99  =  22 ; 
and  for    a;"  =   —  2,    the  same  equation^ 

a;2  +  dxt/  =  22, 

gives,     (—  2)2  +  3  X  -  2  X  -  3   =  4  +  18  =   22. 

The  verifications  could  be  made  in  the  same  way  by  em- 
ploying  Equation  (  2  ). 

Note. — In  equations  similar  to  the  above,  we  generally 
find  but  a  single  pair  of  values,  corresponding  to  the  values 
m  this  equation,  of   y'  =   +  3,    and    a;'  =   +  2. 

The  complete  solution  would  give  four  pairs  of  values. 
ar»  -    y2  _   _  9 


I  y2  -  ary    =        5  f 

j  ay  -  y2  =   _  7  J 
*"•    I  y2  4.  a.2  33       85  f 

j  2a^  +  3x7/  =    470  ) 
j  y»    -    ay  =   -  9  f 

j  5ay  -  3y2  =  32  ) 

^-    "1  ar  +  y2  +  3ay  =  VI  f 
11* 


Ana 

\y  = 

z   4. 
=  5. 

Ans 

X    = 

=  6, 

=  7. 

Ana. 

X   — 

'  y  = 

10. 
9 

Ans, 

\y  = 

7. 
1. 

250 


ELEMENTARY       ALGEBRA. 


THIRD. PAETIOILAR   CASES. 


I'yS  Many  other  equations  of  the  second  degree  may  be 
so  transformed,  as  to  be  brought  under  the  rules  of  sohition 
already  given.  The  seven  following  formulas  will  aid  in 
such  transformation. 

(1.) 
'  When  the  sum  and  difference  are  known: 

X  +  y  =  s 
X  —  y  —  d. 

Then,  page  132,  Example  3, 

s  +  d        1,1,  ,•  s  -  d        1         1, 

^  =  -Y-  =  2'  +  7/^     ^nd     2/  =  __  =   -.-  -^ 

(2.) 
When  the  sum  and  product  are  known: 

x-\-    y  =     s (1.) 

xy  =    p (2.) 

a;2  ^-  2xy  -\-  y"^  =     s\    by  squaring  ( 1 ) ; 
4xy  =  4p,  by  mult.  (  2  )  by  4. 

x^  —  2xy  -f  2/2  —  g2  __  4p^    }jj  subtraction. 


But, 
hence 


x  —  y  =   ±  x/s"^  —  4p,    by  ext.  root. 
X  +  y  =  s; 


X  =  -±  -  ^/s^-  4p 


and, 


2/  =   2  "^  2  V«^^- 


175.  What  is  the  first  formula  of  this  article?    What  the  sncond? 
Third?     Fourth?    Fifth?    Sixt):  ?    Seventh? 


8IMDLrANEOU8       EQUATIONS.  251 

(3.) 
When  the  diifercnne  and  product  are  known: 

X-    ij  =z     d (1.) 

^y'==  P (2.) 

a^  —  2xy  +  y^  =  <^,    by  squaring  ( 1  )  • 

Axy  =  4/>,     mult.  (  2  )  by  4. 

sc*  +  2xy  -^  y^  z=  (P  -f  4p,    by  adding. 

X  +  y  =  ±   y/(P  +  4p 

aj  —  y  =  d 


1 


a;        =        ^d  ±  ^  Vo?2  -f  4/). 

4.) 
When  the  sum  of  the  squares  and  product  are  kno^ni . 
a52  +  y2=  ,..(1.)     xy=p,.(2.)    .'.2xy  =  2p..{3.) 
Adding  ( 1 )  and  ( 3  ),  jc^  +  2a;y  +  y^  =  s  -\-  2p; 


hence,  x  -\-  y  =   ±  ys  -\-  2p     (4.) 

Subtracting  (3)  from  ( 1 ),    x^  —  2xy  +  y^  =  s  —2p; 


hence,  a;  —  y  =    ±  -/^  —  2/>    (6.) 

Combining  (4)  and  (5),  a;  =  ^-/«  +  2p  +  |V«  —  2/>, 
an^  y  =  iVs  -h  2p  —  iV*  —  2/). 

When  the  sura  and  sum  of  the  squares  are  known : 

X   -{-  y    =  8        (1.) 

a*  4-  y'  =  «'       (2.) 

ffi*  -I-  2a:y  -f  y^  =  «*     by  squaring  (  I  ) 
2xy  =  8*  -^  8' 

g2  _    g' 

cry   =    — — -    =  p.  (3.) 


252 


ELEMENTAF.  Y       ALGEBRA 


By  putting  xy  =  p^   and  combining  Equations  (1)  and 
(3  ),  by  Formula  (2),  we  find  the  values  of  x  and  y. 

(6.) 
When  the  sum  and  sum  of  the  cubes  are  known  : 

a   +  2/    =  8  ....     (1.) 

jc3  +  2/3  1=   152  .     .     .     .     (2.) 

7^  4-  3a;2y  +  Sicyz  +  y^  -512  by  cubing  ( 1 ). 

Zx^y  +  3jr?/2  =:   360  by  subtraction. 

^xy{x  -[-  y)   =   360  by  factoring. 

3a;y(8)   =  360  from  Equa.  ( 1  ) 
14:xy  =   360 

hence,  xy  —  15  .     .     .     ,     (3.) 

Combining  ( 1 )  and  (  3  ),  we  find  a;  =  5  and  ?/  =  3. 
When  we  have  an  equation  of  the  form, 

(x  +  yY  +  {x  +  y)  =  g- 

Let  ns  assume  a;  4-  y  =  s. 
Then  the  given  equation  becomes, 

z'^  z  =  q;    and     z  =    -  -±  sj q  +  ^• 

1 


^  +  y  = 


vA^ 


1.  Given 


EXAMPLE! 

xz  ^  y'^     ( 1 ) 


r  xz^  y^     {\)\ 

jaj   +  y   +s   r=    7     (2)  [ 
(a;2  -t-  y2  -f  22  =  21     (3)  ) 


to  find  Xj  y,  and  z. 


SIMULTANEOUS      EQUATIONS.  253 

Transposing  y  in  Equation  (  2 ),  we  have, 

a  +  z  =  7  —  y;         ...    (4.) 

then,  squaring  the  members,  we  have, 

jc2  4-  2a;2  +  22=  49  —  14y  -f  y"^. 

If  now  we  substitute  for  2iBz,  its  value  taken  from  £qua^ 
lion  (  1 ),  we  have, 

a^^   +    2?/2    -I-    22    _     49    _     14y    ^    y2  . 

and  cancelling  y^^  in  each  member,  there  results, 
a^  +  2/^  -f  2^  =  49  —  14  y. 

But,  from  Equation  ( 3  ),  we  see  that  each  member  of  the 
last  equation  is  equal  to  21 ;  hence. 


49 

-14y 

=   21, 

and. 

14y  = 

49  — 

21    =   28, 

hence. 

y 

28 
~   14 

=   2. 

Substituting 

this  value 

of  y  in  Equation  (1), 

gives. 

JC2  =  4  ; 
and  substitutmg  it  in  Equation  ( 4 ),  gives, 

jc  -f  z  =  5,     or    X  z=  b  —  z. 

Substituting  this  value  of  a,  m  the  previous  equation,  wo 

obtain 

52  -  z2  =   4,     or     2^  —  52  =    —  4  ; 

and  by  completmg  the  square,  we  have, 

2^  -  52  +  6.25    =   2.25. 

anrl,       2  —  2.5   =    ±'^2^b=   -\-  1.5,     or     —  1.5 

hence,  z  =   2.5  -f  1.5  =  4,    and    2  =    -f-  2.6  —  1.5   =   1 


254  ELEMENTARY       ALGEBRA. 

2.  Given    x    +  ^/xy  -{-  y    =    ^^  \   xq  find  3.  and 
and    a;2  _j_      xy  -{-  y"^  —  l^Z  )  ^         ^* 

Dividing  the  second  equation  by  the  first,  we  have, 
x  —  ^/xy  -{-    y  =     1 
but,  X  -\-  -/xy  +    y  r=   19 

hence,  by  addition,  2x  -{-  2y  =  26 

or,  X  -{-    y  =   IS 

and  substituting  in  1st  Equa.,  \/xy  +  13   =  19 


or,  by  transposing, 

^^  z=     6 

and  by  squaring. 

xy   =z   36. 

Equation  2d,  is 

a;2  4.  jcy  +  2/2  _   133 

and  from  the  last,  we  have. 

.ixy             =108 

Subtracting, 

a;2  —  2xy  -f  y'^  =     25 

hence. 

X  —  y  z=  ±5 

hut. 

X  -^  y  =     13 

hence,      a;  =  9    and    4 ; 

and 

y  =z  4    and    9. 

PROBLEIMS. 

1.  Find  two  numbers,  such  that  their  sum  shall  bo  15  and 
the  sum  of  their  squares  113. 

Let  X  and  y  denote  the  numbers ;  then, 

a;  -f-  2/  =   15,     (1.)         and     x^  -i-  y^  =  113.     (2.) 

Front  Equation  (  1 ),  we  have, 

a;2  =  225  —  SOy  -f  y^ 

Substituting  this  value  in  Equation  (  2  ), 

225  —  dOy  +  y^  -^  y^  =  113; 


PR0WLEM8.  255 

hence,  2y2  -  30y  =   -  112, 

2/2  —  loy  =   —    50, 

hence,  y'  =  8,     and    y"  —   7. 

Tlio  first  value  of  y  being  substituted  in  Equation  ( 1 ), 
gives  jc'  =  7  ;  and  the  second,  x"  =  8.  Hence,  the  nunt* 
here  are  7  and  8. 

2.  To  find  two  numbers,  such  tliat  their  [»roduct  added  to 
their  sum  shall  be  17,  and  their  sum  taken  from  the  sum  of 
their  squares  shall  leave  22. 

Let  X  and  y  denote  the  nmnbere;  then,  from  the  con- 
ditions, 

(X  -\-  y)  +  xy  =   11.        ...     (1.) 

x'  +  y^-  (x  +  y)    =  22,        ...     (2.) 
Multiplying  Equation  ( 1 )  by  2,  we  have, 

2xy  -f  2(x  +  2/)   =  34.        ...     (3.) 
Adding  (  2  )  and  {  3  ),  we  have, 

x'^  2xy  +  y^-\-  (x-\-  y)  =  56 ; 

hence,  {x  +  yY  -\-  (x  -{-  y)  =  56.      .     .     (4.) 

Regarding  (x  +  y)  as  a  single  unknown  quantity  (page 
248), 

x  +  y  =   -l±  y/sT  +  \  =  1. 

Substituting  this  value  in  Equation  ( 1 ),  we  have, 

T  -\-  xy  =   17,      and     y  —  5, 
Hence,  the  numbers  are  2  and  5. 

3.  Wliat  two  numbers  are  those  whose  sum  is  8,  and  suni 
of  Uieir  squares  34  ?  Ans,   6  and  3. 


256  ELEMENTARY      ALGEBRA. 

4.  It  is  required  to  find  two  such  numbers,  tLat  the  first 
shall  be  to  the  second  as  the  second  is  to  16,  and  the  sum  ol 
whose  squares  shall  be  225  ?  Ans.    9  and  12. 

5.  What  two  numbers  are  those  which  are  to  each  other 
as  3  to  5,  and  whose  squares  added  together  make  1666  ? 

Ans.    21  and  35. 

6.  There  are  two  numbers  whose  difference  is  7,  and  half 
their  product  plus  30  is  equal  to  the  square  of  the  less 
number:  what  are  the  numbers?  A7is.    12  and  19. 

7.  What  two  numbers  are  those  Avhose  sum  is  5,  and  the 
pum  of  their  cubes  35  ?  Ans.    2  and  3. 

8.  What  two  numbers  are  those  whose  sum  is  to  the 
greater  as  11  to  7,  and  the  difference  of  whose  squares  is 
132  ?  Ans.    14  and  8. 

9.  Divide  the  number  100  into  two  such  parts,  that  the 
product  may  be  to  the  sum  of  their  squares  as  6  to  13. 

Ans.    40  and  60 

10.  Two  persons,  A  and  j5,  departed  from  different  places 
at  the  same  time,  and  traveled  towards  each  other.  On 
meeting,  it  appeared  that  A  had  traveled  18  miles  more 
than  i? ;  and  that  A  could  have  gone  J3''s  journey  in  15f 
days,  but  J^  would  have  been  28  days  in  performing  A'>s 
journey  ;  how  far  did  each  travel  ?  .        \  A,  72  miles. 


Ans.  j 


J?,  54  miles. 

11.  There  are  two  numbers  whose  difference  is  15,  and 
half  their  product  is  equal  to  the  cube  of  the  leeser  number : 
wliat  are  those  numbers?  A?2S.    3  and  18. 

12.  What  two  numbers  are  those  whose  sum,  multiplied 
'^by  the  greater,  is  equal  to  77  ;  and  whose  difference,  multi- 
plied by  the  less,  is  equal  to  12  ?  _ 

Ans.    4  and  7,  or  1^2  and  V  \/2. 


PROBLEMS.  257 

13.  Divide  100  into  two  such  parts,  that  the  sum  of  their 
square  roots  may  be  14.  Ans.   64  and  36. 

14.  It  is  required  to  divide  the  number  24  into  two  such 
parts,  that  their  product  may  be  equal  to  35  times  their  dif. 
ference.  Ans,    10  and  14, 

15.  The  sum  of  two  numbers  is  8,  and  the  sum  of  tlioir 
cubes  is  152  :  what  are  the  numl)ers  ?  Ans.   3  and  5. 

16.  Two  merchants  each  sold  the  same  kind  of  stuff;  the 
second  sold  3  yards  more  of  it  than  the  first,  and  together 
they  receive  35  dollars.  The  first  said  to  the  second,  "I 
would  have  received  24  dollars  for  your  stuff;"  the  other 
replied,  "And  I  should  have  received  12^  dollars  for  yours :" 
how  many  yards  did  each  of  them  sell  ? 

.        \  1st  merchant  a'  =  15,  a;"  =  5. 

^"*-J2d  "         y'  =  18,     "'■'     y"=8. 

17.  A  widow  possessed  13,000  dollars,  which  she  divided 
into  two  parts,  and  placed  them  at  interest  in  such  a  manner 
that  the  incomes  from  them  were  equal.  If  she  had  put  out 
the  first  portion  at  the  same  rate  as  the  second,  she  "would 
have  drawn  for  this  part  360  dollars  interest ;  and  if  she 
had  placed  the  second  out  at  the  same  rate  as  the  first,  she 
would  have  drawn  for  it  490  dollars  interest :  what  were 
the  two  rates  of  interest  ?  Ans.   7  and  6  per  cent. 

18.  Find  three  numbers,  such  that  the  difference  between 
the  third  and  second  shall  exceed  the  difference  between  the 
second  and  first  by  6  ;  that  the  sum  of  the  numbers  shall  be 
33,  and  the  sum  of  their  squares  467. 

Ans.   5,  9,  and  19. 

19.  What  number  is  that  which,  being  divided  by  the 
product  of  its  two  digits,  the  quotient  will  be  3 ;  and  if  18 
be  added  to  it,  the  resultmg  number  will  be  expressed  by 
the  digits  inverted  ?  A?is.   24. 


ELEMENTARY        ALGEBliA. 


20.  What  two  numbers  are  those  which  are  to  each  othei 
as  W2  to  w,  and  the  sum  of  whose  squares  is  5  ? 

m^  n-y/b 


Ans, 


^/m?  + 


m^  -}-  n^ 


21.  What  two  numbers  are  those  which  are  to  each  oilier 
as  m  to  71,  and  the  difference  of  whose  squares  is  6  ? 


Ans. 


-v/m^ 


i^      y/m^  —  w^ 


22.  Required  to  find  three  numbers,  such  that  the  product 
of  the  first  and  second  shall  be  equal  to  2  ;  the  product  of 
the  first  and  third  equal  to  4,  and  the  sum  of  the  squares 
of  the  second  and  third  equal  to  20.  Ans.    1,  2,  and  4. 

23.  It  is  requfred  to  find  three  numbers,  whose  sum  shall 
be  38,  the  sum  of  their  squares  634,  and  the  difference 
betw^een  the  second  and  first  greater  by  7  than  the  difference 
between  the  third  and  second.  Ans,    3,  15,  and  20. 

24.  Required  to  find  three  numbers,  such  that  the  product 
of  the  first  and  second  shall  be  equal  to  a ;  the  product  of 
the  first  and  third  equal  to  h ;  and  the  sum  of  the  squares 
of  the  second  and  third  equal  to  c. 


Ans. 


X    — 


y 


R 

+   62 

G 

^''s/a^ 

c 

^Va2  4.  52 


25.  What  two  numbers  are  those,  whose  sum,  multiplied 
by  the  greater,  gives  144 ;  and  whose  difference,  multiplied 
by  the  less,  gives  14  ?  Ans,   9  and  1. 


PBOPOBTIONS      AND      PROGRESSIONS.      259 


CHAPTER  IX. 

OF     PROPORTIONS     AND     PROGRESSIONS. 

I T6.  IHvo  quantities  of  the  same  kind  may  be  compared, 
the  one  with  the  other,  in  two  ways : 

Ist.  By  considering  how  much  one  is  greater  or  less  than 
the  Other,  which  is  shown  by  their  difference ;  and, 

2d.  By  considering  how  many  times  one  is  greater  or  less 
than  the  other,  which  is  showTi  by  their  quotient. 

Tims,  in  comparing  the  numbers  3  and  12  together,  with 
respect  to  their  difference,  we  find  that  12  exceeds  3,  by  9 ; 
and  in  comparing  them  together  wuth  respect  to  their  quo- 
tient, we  find  that  12  contains  3,  four  times,  or  that  12  is  4 
times  as  great  as  3. 

Tlie  first  of  these  methods  of  comparison  is  called  Arith- 
metical Proportion^  and  the  second,  Geometrical  Fropor- 
lion. 

Hence,  Arithmetical  Proportion  considers  the  relation  of 
quantities  with  respect  to  tJieir  difference^  and  Geometrical 
Proportion  the  relation  of  quantities  with  respect  to  their 
quotient. 

176.  In  how  many  ways  may  two  quantities  be  compared  the  one  with 
the  other?  What  does  the  first  method  consider?  What  the  seconl? 
Wliat  is  the  first  of  these  methods  called  ?  What  b  the  second  called  P 
How  then  do  you  define  the  two  proportions  ? 


260  ELEMENTARY      ALGEBEA. 

OF   AEITHMETICAI.   PEOPOETION   AOT)   PKOGEESSION. 

I'yy.  If  we  have  four  numbers,  2,  4,  8,  and  10,  of  which 
tlie  difference  between  the  first  and  second  is  equal  to 
the  difference  between  the  third  and  fourth,  these  numbers 
are  said  to  be  in  arithmetical  proportion.  The  first  term  15 
is  called  an  antecedent^  and  the  second  term  4,  with  wliich 
it  is  compared,  a  consequent.  The  number  8  is  also  called 
an  antecedent,  and  the  number  10,  with  which  it  is  com-* 
pared,  a  consequent. 

When  the  difference  between  the  first  and  second  is  equal 
to  the  difference  between  the  third  and  fourth,  the  four 
numbers  are  said  to  be  in  proportion.     Thus,  the  numbers, 
2,     4,     8,     10, 

are  in  arithmetical  proportion. 

ITS.  When  the  difference  between  the  first  antecedent 
and  consequent  is  the  same  as  between  any  two  consecutive 
terms  of  the  proportion,  the  proportion  is  called  an  arith' 
metical  progression.  Hence,  a  progression  by  differences^ 
or  an  arithmetical  pyrogression.,  is  a  series  in  which  the  suc- 
cessive terms  are  continually  increased  or  decreased  by  a 
constant  number,  which  is  called  the  common  difference  of 
the  progression. 

Thus,  in  the  two  series, 

1,       4,       7,      10,     13,     16,     19,     22,     25,  .  .  . 
60,     56,     52,     48,     44,     40,     36,     32,     28,  .  .  . 

177.  "When  are  four  numbers  in  arithmetical  proportion  ?  "What  is  the 
first  called?  What  is  the  second  called?  What  is  the  third  called? 
What  is  the  fourth  called  ? 

178.  "What  is  an  arithmetical  progression  ?  "What  is  the  number  called 
by  which  the  terms  are  increased  or  diminished  ?  What  Is  an  increasing 
progression  ?  What  is  a  decreasing  progression  ?  Which  term  is  only 
an  antecedent  ?    "Which  only  a  consequent  ? 


AKI'lUMBTICAL      PKOOKK88ION.  261 

the  first  is  called  an  increasing  progression^  of  which  tho 
common  difference  is  3,  and  tke  second,  a  decreasing  pro- 
ffressioti,  of  which  the  common  difference  is  4. 

In  general,  let  a,  ^,  c,  c/,  e,  /",  ...  denote  the  terms  of 
a  progression  by  differences ;  it  has  been  agreed  to  write 
them  thus : 

a,b,c,d.e,/.g.h,i.k... 

This  series  is  read,  a  is  to  J,  as  5  is  to  c,  as  c  is  to  (7,  as  df 
is  to  e,  <fcc.  This  is  a  series  of  continued  equi-differences^  in 
which  each  term  is  at  the  same  time  an  antecedent  and  a 
consequent,  with  the  exception  of  the  first  term,  which  is 
only  an  antecedent^  and  the  last,  which  is  only  a  consequent, 

179.  Let  d  denote  the  common  difference  of  the  pro- 
giesion, 

a  ,  h  .  c  ,  e  .  f  ,  g  .  h.  <fec., 

which  we  will  consider  increasing. 

From  the  definition  of  the  progression,  it  evidently  fol 
lows  that, 

h  =  a  +  d^    c  =  b  +  d  =  a  +  2d^    e  =  c-\-d=a-\-  Sd; 

and,  in  general,  any  term  of  the  series  is  equal  to  the  first 
tenn,  plus  as  many  times  the  common  difference  as  there  are 
preceding  terms. 

Thus,  let  /  be  any  term,  and  n  the  number  which  marks 
the  place  of  it ;  the  expression  for  this  general  term  is, 

I  =z  a  -\-  {n  —  \)d. 

Hence,  for  finding  the  last  term,  we  have  the  following 


179.  Give  the  rule  for  finding  the  last  term  of  a  ecries  when  the  pro 
^reesion  is  Increasing. 


262  ELEMENTARY      ALGEBKA, 


RULE. 

I.  Multiply  the  common^ difference  by  the  number  of 
terms  Less  one: 

II.  To  the  x>roduct  a!Ud  the  first  term  ;  the  sum  will  be 
the  last  term. 

EXAMPLES. 

The  formula, 

I  —  a  -\-  {n  —  \)d^ 

serves  to  find  any  term  whatever,  without  determiningr  all 
those  Avhich  precede  it. 

"** 

1.  If  we  make    7i  =  1,    we  have,    I  =z  a;    that  is,  the 

series  will  have  but  one  term. 

2.  If  we  make  n  —  2^  we  have,  I  =  a  -\-  d\  that  is, 
the  series  will  have  two  terms,  and  the  second  term  is  equal 
to  the  first,  plus  the  common  difierence. 

3.  If   a  =  3,    and    d  =z  2^    what  is  the  3d  term? 

Ans.    1. 

4.  If   a  =  5,    and    c?  =  4,    what  is  the  6th  term? 

A92S.    25 

5.  If   a  =  7,    and    d  —  5,    what  is  the  9th  term? 

Ans.    47. 

6.  If   a  =   8,  and    c?  =  5,    what  is  the  10th  term  ? 

A71S.   5'i 

7.  If   «  =   20,    and    d  =  4,    what    is  the  12th  term? 

Ans.   64, 

8.  If   a  =  40,    and    d  =  20,    what  is  the  50th  teim? 

A71S.    1020. 

9.  If   a  =  45,    and    d  =  30,    what  is  the  40th  term  ? 

Ans.    1215. 


ARITHMETICAL      PKOOKKSSION  203 

10.  If   a  =  30,    and    d  =  20,    what  is  the  60th  term? 

Atis.    1210. 

11.  ir   a  —  50,    and    d  =   10,    Mli:it  is  the  100th  tcrni? 

A/ts.    3  010. 

12.  To  find  tlie  50th  tenn  of  tlie  jjrugression, 

1   .  4  .  7  .  10  .  13  .  16  .  19 

we  have,  ^  =  1  +  40  x  3  =   148. 

13.  To  find  the  60th  temi  of  the  progression, 
1   .  5  .  9  .   13  .   17  .  21   .  25  .  .  . 
we  have,  /  =   1  +  59  x  4  =   237. 

180.     It'   the   progression   were  a  decreasing  one,  we 

should  !i:ive, 

/   =r  a  —  {n  —  l)d. 

Ileim,  t(i  ilnd  the  last  term  of  a  decreasing  progression,  we 
have  tiie  followmg 

RULE. 

L  Multiple/  the  common  difference  by  the  number  of  terms 
less  one : 

n.  Subtract  the  product  from.  Vie  first  term ;  the  ro- 
mainder  will  be  the  last  term. 

EXAMPLES. 

1.  The  first  tenn  of  a  decreasing  progression  is  60,  the 
number  of  terms  20,  and  the  common  difierence  3  :  wliat  is 
the  last  term? 

l=za-(n-\)d,   gives  i  =  60  -  (20  — 1)3  =  60 -57  =  3 

180.  Give  the  rule  for  finding  the  laat  term  of  a  series,  nhtn  the  pro 
grcssion  is  decreasing. 


264  ELEMENTARY       ALGEBRA. 

2.  The  first  term  is  90,  the  common  difference  4,  and  the 
number  of  terms  15  :  what  is  the  last  term  ?  Ans.   34. 

3.  The  first  term  is  IGO,  the  number  of  terms  40,  and  the 
common  difference  2  :  what  is  the  last  term  ?  Ans.   22. 

4.  The  first  term  is  80,  the  number  of  terms  10,  and  the 
common  difference  4  :  what  is  the  last  term  ?  Ans.   44. 

5.  The  first  term  is  600,  the  number  of  terms  100,  and 
the  common  difference  5  :  what  is  the  last  term  ? 

Ans.    105. 

6.  The  first  term  is  800,  the  number  of  terms  200,  and 
the  common  difference  2  :  what  is  the  last  term  ? 

Ans.   402. 

1§1.  A  progression  by  differences  being  given,  it  is  pro- 
posed to  prove  that,  the  sum  of  any  two  terms^  taken  at 
equal  distances  from  the  two  extremes^  is  equal  to  the  sum 
of  the  two  extremes. 

That  is,  if  we  have  the  progression, 

2  .  4  .  6  .  8  .  10  .  12, 
we  wish  to  prove  generally,  that, 

4  +  10,     or     6  +  8, 
is  equal  to  the  sum  of  the  two  extremes,  2  and  12. 

Let  a.b.c,e.f...  i  .  k  .  I,  be  the  proposed 
progression,  and  n  the  number  of  terms. 

We  will  first  observe  that,  if  x  denotes  a  term  which  has 
p  terms  before  it,  and  y  a  term  which  has  p  terms  after  it, 
we  have,  from  what  has  been  said, 

181.  In  every  progression  by  differences,  what  is  the  sum  of  the  two 
extremes  equal  to?  What  is  the  rule  for  finding  the  sum  of  an  arith- 
tnctical  series? 


ABITHMETIOAL      PBOOBEBSION.  2G5 

X  =  a  -\-  p  X  d, 
and,  y  =  '  —  p  X  d; 

whence,  by  addition,    x  +  y  =  a  +  I, 
which  proves  the  proposition. 

Referring  to  the  previous  example,  if  we  suppose,  in  the 
first  place,  x  to  denote  the  second  terra  4,  then  y  will  de- 
note the  term  10,  next  to  the  last.  If  x  denotes  the  third 
term  6,  then  y  will  denote  8,  the  third  term  from  the  last. 

To  apply  this  principle  in  findmg  the  sum  of  the  tei-ms 
of  a  progression,  WTite  the  terms,  as  below,  and  then 
again,  in  an  inverse  order,  viz. : 

I   .  k  .  i c  ,  b  ,  a. 

Callmg  8  the  sum  of  the  terms  of  the  first  progression, 
IS  will  be  the  sum  of  the  terms  of  both  progressions,  and 
we  shall  have, 

2S={a+t)  +  (h+k)-^{c+i)  .  .  .  +(i-hc)+(A;+*)  +  (^+«). 

Now,  since  all  the  parts,  a  -^r  I,  h  -\-  k,  c  -{-  i  ,  .  ,  ^re 
equal  to  each  other,  and  their  number  equal  to  w, 

28=  (a  +  l)  X  w,    or     S  -  y-^)  X  w- 

Ilonce,  for  finding  the  sum  of  an  arithmetical  series,  we 
have  the  following 

RULE. 

L  Add  the  two  extremes  together^  and  take  haXfth^ir  sum : 
U.  Mtdtiply  this  half -sum  by  the  number  of  terms  ;  the 
product  wiU  be  the  sum  of  the  series. 
12 


266  ELEMEI?rTAEY       ALGEBRA, 


EXAMPLES. 

1.  The  extremes  are  2  and  16,  and  the  number  of  terms 
8  :  what  is  the  sum  of  th.e  series  ? 


S  =  [—^)  X  ^^:    gives     S  =  -— ^  X  8 


Y2. 


2.  The  extremes  are  3  and  27,  and  the  number  of  tenna 
12  :  what  is  the  sum  of  the  series  ?  Ans,   180. 

3.  The  extremes  are  4  ard  20,  and  the  number  of  terms 
10:  what  is  the  sum  of  the  series?  Ans.    120. 

4.  The  extremes  are  100  and  200,  and  the  number  of 
terms  80 :  what  is  the  sum  of  the  series  r  A7is.    12000. 

5.  The  extremes  are  500  and  60,  and  the  number  of  terms 
20  :  what  is  the  sum  of  the  series  ?  Ans.   5600 

6.  The  extremes  are  800  and  1200,  and  the  number  of 
terms  50 :  what  is  the  sum  of  the  series  ?  Ans.   60000. 

1 82.  In  arithmetical  proportion  there  are  five  menibers 
to  be  considered : 

1st.   The  first  term,  a. 

2d.    The  common  difierence,  d, 

3d.    The  number  of  terms,  n. 

4th.  The  last  term,  I. 

6th.  The  sum,  S. 

The  formulas, 

I  =  a  -^  {n  —  l)d,     and     >S'  =  | — - — J  x  n, 

contain  five  quantities,  a,  d,  w,  I,  and  S,  and  consequently 
give  rise  to  the  following  general  problem,  viz. :  Any  three 

162.  How  many  numbers  are  considered  in  arithmetical  proportion? 
What  are  they  ?  In  every  arithmetical  progression,  what  is  the  common 
difference  equal  to  ? 


AUITHMETIOAL      PROOEESBION.  267 

of  these  five  quantities  being  given,  to  determine  the  other 
tioo. 

We  already  know  the  value  of  aS'  in  terms  of  a,  w,  and  L 
From  the  formula, 

I  =  a  -^  (n  —  1)(7, 
we  find,  a  =    I  —  {n  —  \)d. 

That  is :  The  first  term  of  an  increasing  arithmetical  pro- 
gression is  equal  to  the  last  term,  minus  the  product  of  the 
common  difference  by  the  number  of  terms  less  one. 

From  the  same  formula,  we  also  find, 

I  —  a 


d  = 


n 


Tliat  is :  In  any  ai'ithmetical  progression,  the  common  dif- 
ference is  equal  to  the  last  term,  minus  the  first  term,  divided 
by  the  numbar  of  terms  less  one. 

The  last  term  is  16,  the  first  tcnn  4,  and  the  number  of 
tenns  6 ;  what  is  the  common  difierence  ? 

Tlie  fonntda,  d  = 

^  n  —  1 

7        16-4 
gives,  d  =  — ~ —  =  3. 

2.  The  last  tenn  is  22,  the  first  term  4,  and  the  number 
of  terms  10 :  what  is  the  common  difference?  Ans.   2. 

183.  The  last  principle  afibrds  a  solution  to  the  follow- 
Ing  question : 

To  find  a  number  m  of  arithmetical  means  between  two 
given  numbers  a  and  b. 


1S8.  How  do  jou  find  any  number  of  ftrithroctical  means  betweci  two 
0\ien  n'unbere  ? 


268  ELEAtENTAIiY      ALGEBRA. 

To  resolve  this  question,  it  is  first  necessary  to  find  the 
common  difierence.  Now,  we  may  regard  a  as  the  first 
term  of  an  arithmetical  progression,  h  as  the  last  term,  and 
the  required  means  as  intermediate  terms.  The  number  of 
terms  of  this  progression  will  be  expressed  by  m  4-  2. 

Now,  by  substituting  in  the  above  formula,  h  for  Z,  and 
>?i  -f  2  for  ?2,  it  becomes, 

,_        h  —  a        _    ^  ~  ^  . 

~  7)1  -{-  2  —  \   ~  m  +  1  ' 

that  is :  The  common  difference  of  the  required  progression 
is  obtained  hy  dividing  the  difference  between  the  given 
numbers^  a  and  b,  by  the  required  number  of  means  plus  one. 

Having  obtained  the  common  difference,  d^  form  the  second 
term  of  the  progression,  or  the  first  arithmetical  mean^  by 
adding  d  to  the  fii?st  term  a.  The  second  mean  is  obtained 
by  augmenting  the  first  mean  by  c?,  &c. 

1.  Find  three  aritlimetical  means  between  the  extremes 
2  and  18. 

b  —  a 


The  formula,  d 


m  +  l' 


^         18-2 
gives,  d  =  — —    =4', 

hence,  the  progression  is, 

2  .  6  .  10  .  14  .  18. 

2.  Find  twelve  arithmetical  means  between  12  and  77. 

b  —  a 


Tlie  formula,  d  = 


m  +  1  ' 


^        77-12 
gives,  d  =^ —    =  5  ; 

hence,  the  progression  is, 

12  .  17  .  22  .  27      ...  77. 


ARITHMETICAL       PBOORE88ION.  2(>9 

184.  Remark. — If  the  same  number  of  arithmetical 
means  are  inserted  between  all  the  terms,  taken  two  and 
two,  these  terms,  and  the  arithmetical  means  united,  will 
form  one  and  the  same  progression. 

For,  let  a.b.c,e.f,,'.  be  the  proposed  progression 
and  m  the  number  of  means  to  be  inserted  between  a  and 
i,  b  and  c,  c  and  e  .  .  .  .  &c. 

From  what  has  just  been  said,  the  common  difference  of 
each  partial  progression  will  be  expressed  by 

b  —  a       c  —  b       e  —  c 

m  +  1 '    wi  +  1 '    m  4-  1  *  *  ' 

expressions  which  are  equal  to  each  other,  since  a,  5,  c  .  .  . 
are  in  progression  ;  therefore,  the  common  difference  is  the 
same  in  each  of  the  partial  progressions  ;  and,  since  the  last 
term  of  the  first  fomis  the  Jirst  term  of  the  second,  &c.,  we 
may  conclude,  that  all  of  these  partial  progressions  form  a 
single  progression. 

EXAMPLES. 

1.  Find  the  sum  of  the  first  fifty  terms  of  the  progression 
2  .  9  .  16  .  23  .  .. 

For  the  60th  term,  we  have, 

/  =   2  4-  49  X  7   =   345. 

50 
Hence,     S  =   (2  -\-  345)  x  y  =  347  X  25  =  8676. 

2.  Find  the  100th  term  of  the  series  2  .  9  .  16  .  23  .  .  . 

A?is,   695 

3.  Find  the  sum  of  100  terms  of  the  series  1.3.5.7. 
9 A71S    10000. 


270  ELEMENTARY       ALGEBRA. 

4.  The  greatest  term  is  10,  the  common  difference  3,  and 
the  number  of  terms  21 :  what  is  the  least  term  and  tiie 
fiimi  of  the  series  ? 

A71S.   Least  term,  10  ;  sum  of  series,  840. 

5.  The  first  term  is  4,  the  common  difference  8,  and  the 
number  of  terms  8 :  what  is  the  last  term,  and  the  sum  of 
the  series  ?  Ans.  j  Last  term,    60 

(  Sum        =  256. 

6.  The  first  term  is  2,  the  last  term  20,  and  the  number 
of  terms  10 :  what  is  the  common  difference  ?  Atis.    2. 

7.  Insert  four  means  between  the  two  numbers  4  and  19 : 
what  is  the  series  ?  Ans.   4  .  7  .  10  .  13  .  16  .  19. 

8.  The  first  term  of  a  decreasing  arithmetical  progression 
is  10,  the  common  difference  one-third,  and  the  number  of 
terms  21 :  required  the  sum  of  the  series.  Ans.    140. 

9.  In  a  progression  by  differences,  having  given  the  com- 
mon difference  6,  the  last  term  185,  and  the  sum  of  the 

/*^   terms  2945  :  find  the  first  term,  and  the  number  of  terms. 
Ans.   First  term  =z  5  ;  number  of  terms,  31. 

10.  Find  nine  arithmetical  means  between  each  antecedent 
and  consequent  of  the  progression  2. 5. 8. 11. 14... 

Ans.   Common  diff.,  or  d  =  0.3. 

11.  Find  the  number  of  men  contained  in  a  triangular 
battalion,  the  first  rank  containing  one  man,  the  second  2, 
the  third  3,  and  so  on  to  the  n^^,  which  contains  n.  In  other 
words,  find  the  compression  for  the  sum  of  the  natural  num- 
bers 1,  2,  3  .  .  .,  from  1  to  w  inclusively. 

Ans.   S  =  ^^t.1) 
2 

12.  Find  the  sum  of  the  n  first  terms  of  the  progression 
of  uneven  numbers,  1.3.6.7.9,...  Ans.   JS  —  «^ 


GEOMETRICAL     PBOrOBTION.  271 

18.  One  hundred  stones  being  placed  on  the  ground  in  a 
straight  line,  at  the  distance  of  2  yards  apart,  how  flir  will 
a  person  travel  who  shall  bring  them  one  by  one  to  a  basket, 
placed  at  a  distance  of  2  yards  from  the  first  stoiu-  *.* 

Ans,   11  mile-',  SIO  yards. 


GEO^VIETRICAL    PROPORTION    A^T>     I'ilOORESSION. 

185.  liatio  is  the  quotient  arising  from  dividing  one 
quantity  by  another  quantity  of  the  same  kind,  regarded 
as  a  standard.  Thus,  if  the  numbers  3  and  6  have  the  same 
unit,  the  ratio  of  3  to  6  will  be  expressed  by 

I"- 

And  m  general,  if  A  and  J^  represent  quantities  of  the  same 
kind,  the  ratio  of  ^  to  i?  will  be  expressed  by 

B 

a' 

186.  The  character  oc  indicates  that  one  quantity  is 
proportional  to  another.    Thus, 

A  oc  B, 
is  read,  A  proportional  to  B. 

If  there  be  four  numbers, 

2,     4,     8,     16, 

having  such  values  that  the  second  di^^ded  by  the  first  is 
equal  to  the  fourth  divided  by  the  third,  the  numbers  are 

185.  What  is  ratio  ?    What  is  the  Mtio  of  3  to  6  ?     Of  4  to  12  ♦ 

186.  What  is  proportion?  How  do  you  express  that  four  numbers 
arc  in  proportion  ?  What  are  the  numbers  called  ?  What  are  the  firsi 
and  fourth  terras  c\Iled  ?    What  the  second  and  third  ? 


272  ELEMENTARY       ALGEBRA 

said  to  form  a  proportion.  And  in  general,  if  there  be  foni 
quantities,  A,  J3,  (7,  and  J9,  having  such  values  that, 

:?  _  :? 

A  ~   C 

then,  A  is  said  to  have  the  same^ratio  to  J5  that  C  has  to  Di 
or,  the  ratio  of  A  to  J5  is  equal  to  the  ratio  of  C  to  D. 
When  four  quantities  have  this  relation  to  each  other,  com* 
pared  together  two  and  two,  they  are  said  to  form  a  geo- 
metrical proportion. 

To  express  that  the  ratio  of  ^  to  J5  is  equal  to  the  ratio 
of  C  to  J9,  we  write  the  quantities  thus, 

A  :  J^  ::   G  :  D; 

and  read,  ^  is  to  .B  as  C  to  D. 

The  quantities  which  are  compared,  the  one  with  the 
other,  are  called  terms  of  the  i:>roportion.  The  first  and  last 
terms  are  called  the  two  extremes^  and  the  second  and  third 
terms,  the  two  means.  Thus,  A  and  D  are  the  extremes, 
and  _Z?  and  C  the  means. 

1§7.  Of  four  terms  of  a  proportion,  the  first  and  third 
are  called  the  antecedents^  and  the  second  and  fourth  the 
consequents  ;  and  the  last  is  s;i)il  to  be  a  fourth  proportional 
to  the  other  three,  taken  in  order.  Thus,  in  the  last  pro- 
portion A  and  G  are  the  antecedents,  and  B  and  D  the  con- 
sequents. 

I  §8.  Three  quantities  are  in  proportion,  when  the  first 
has  the  same  ratio  to  the  second  that  the  second  has  to  the 

187.  In  four  proportional  quantities,  what  are  the  first  and  third  called  ? 
What  the  second  and  fourth  ? 

188.  When  are  three  quantities  proportional?  What  is  the  middle  one 
called  f 


OKOMETEICAL      PEOPOBTION.  273 

third  ;  and  then  the  middle  term  is  said  to  be  a  mean  pro- 
portional between  the  other  two.     For  example, 

8  :  6  :;  6  ;  12;  /  IM^ 

and  6  is  a  mean  proportional  between  3  and  12. 

18!>.  Four  quantities  are  said  to  be  in  proportion  by  in- 
version^  or  mveisely^  when  the  consequents  are  made  the 
antecedents,  and  the  antecedents  the  consequents. 

Thus,  if  we  have  the  proportion, 

3  :  6  :  :  8  •  16, 
the  inverse  proportion  would  be, 

6  :  3  :  :  16  .  8. 

190.  Quantities  are  said  to  be  in  proportion  by  altema-  ^Ay^ 
tion^  or  alternately^  when  antecedent  is  compared  with  ante- 
cedent, and  consequent  with  consequent. 

Thus,  if  we  have  the  proportion, 

3  :  6  :  :  8  :  16, 

the  alternate  proportion  would  be, 

3  :  8  :  :  6  :  16. 

191.  Quantities  are  said  to  be  in  proportion  by  contpo 
eition^  when  the  sum  of  the  antecedent  and  consequent  if 
compared  either  with  antecedent  or  consequent  . 

Thus,  if  we  have  the  proportion,  ^^^^ 

2  :  4  ;  :  8  :  16, 

189.  When  are  quantities  said  to  be  in  proportion  by  inversion,  or  to 
versely  ? 

190.  When  arc  quantities  in  proportion  by  alternation? 

191.  When  are  quantities  in  proportion  by  composition? 

12* 


274  ELEMENTARY      ALGEBRA. 

the  proportion  by  composition  would  be, 

2  +  4  :  2  :  :  8  +  16  :  8; 
and,  2  +  4  :  4  :  :  8  -h  16  :  16. 

192.  Quantities  are  said  to  be  in  proportion  by  division, 
when  the  difference  of  the  antecedent  and  consequent  it^ 
compared  either  with  antecedent  or  consequent. 

Thus,  if  we  have  the  proportion, 

3  :  9  :  :  12  :  36, 
the  proportion  by  division  will  be, 

9  —  3  :  3  :  :  36  --  12  :  12; 
and,  9  —  3  :  9  :  :  36  —  12  :  36 ; 

193.  Equi-multiples  of  two  or  more  quantities  are  the 
products  which  arise  from  multiplying  the  quantities  by  the 
same  number. 

Thus,  if  we  have  any  two  numbers,  as  6  and  6,  and  mul- 
tiply them  both  by  any  number,  as  9,  the  equi-mulfiples  -will 
be  54  and  45 ;  for, 

6  X  9   =   54,    and    5  X  9   =  45. 

Also,    m  X  A^    and    m  x  B^    are  equi-multiples  of  A  and 
B^  the  common  multiplier  being  m. 

194.  Two  quantities  A  and  B^  which  may  change  theii 
values,  are  reciprocally  or  inversely  proportional^  when  one 
is  proportional  to  unity  divided  by  the  other ^  and  then  the^ft 
product  remains  constant. 

192.  When  are  quantities  in  proportion  by  division  ? 

1&8.  What  are  equi-jQultipIes  of  two  or  more  quantities  ? 

194.  When  are  two  quantities  said  to  be  reciprocally  proportional  P 


OKOMETRICAL      PROPOBTION.  275 

We  express  this  reciprocal  or  inverse  relation  thus, 

in  which  A  is  said  to  be  inversely  proportional  to  B, 

195.  If  we  have  the  proportion, 

A  :  B  ::   C  :  By 

B        B 
we  have,  -r  =  y,,    (Art.  186); 

and  by  cleaiing  the  equation  of  fractions,  we  have, 
BC  =  AB, 

That  is :  Of  four  proportional  quantities^  the  product  of 
the  two  extremes  is  equal  to  th^  product  of  the  two  means. 

This  general  principle  is  apparent  in  the  proportion  be- 
tween the  numbers, 

2  :  10  :  :  12  :  60, 
which  gives,      2  x  60  =  10  x  12  =  120. 

196.  If  four  quantities,  A^  J5,  C,  2>,  are  so  related  to 
each  other,  that 

A  X  B  =z  B  X  C, 

we  shall  also  have,  -j  =  -?^ ; 

and  hence,  A  :  B  i  :  C  i  B, 

That  is :  If  the  product  of  two  quantities  is  equal  to  the 
product  of  two  other  quantities,  two  of  them  may  be  made 
the  extremes^  and  the  other  two  the  means  of  a  proportion. 

196.  If  four  quantitiea  are  proportional,  what  is  the  product  of  the  two 
means  equal  to  ? 

196.  If  the  product  of  two  quantities  is  equal  to  the  product  of  two 
olLcr  quantities,  WAy  the  four  be  placed  in  a  proportion  f    How  ? 


276  ELEMENTARY       ALGEBRA. 

Thus,  if  we  have, 

2X8  =  4X4, 
we  also  have, 

2  :  4  :  :  4  :  8. 

1 97.  If  we  have  tliree  proportional  quantities, 

A  ',  B  :i  B  '.   C, 

B         C 
we  have,  -j  =  -^; 

hence,  B^  =  AC. 

That  is :   If  three  quantities  are  proportional^  the  square  of 
VA^   the  middle  term  is  equal  to  the  product  of  the  two  extremes 

Thus,  if  we  have  the  pro])ortion, 

3  :  6  :  :  6  :  12, 
we  shall  also  have, 

6  X  6   =   62  =   3  X  12   =  36. 

198.  If  we  have, 

A  :  B  :  :   (7  :  X>,    and  consequently,    -^  =  -^, 

c 

multiply  both  members  of  the  last  equation  by    ^,    and 

we  then  obtain, 

C  _  B 
A  ~  B' 

and,  hence,  A  :  C  :  :  B  :  B. 

That  is  :  If  four  quantities  are  proportional^  they  will  he 
in  proportion  by  alternation, 

197.  If  three  quantities  are  proportional,  what  is  the  product  of  the 
jxtremes  equal  to  ? 

198.  If  four  quantities  are  proportional,  will  they  De  in  proportion  by 
alternation  ? 


OEOMETRIOAL      PROPORTION.  277 

Let  116  take,  as  an  example, 

10  :  15  :  :  20      30. 

Wp  shall  have,  by  alternating  the  terms, 

10  :  20  •  :  15  :  '30. 

1 99.  If  we  have, 

A  :  B  ::  C  :  D,    and    A  :  J^  :  :  E  :  T, 

we  shall  also  have, 

A=    C^    and    ^   =^; 

J)        F 

hence,  -^  =  -= ,    and     C  :  D  :  :  E  i  F 

That  is :  If  there  are  ttoo  sets  of  projyortions  having  an  an 
tecedent  and  consequent  in  the  one^  equal  to  an  antecedeiit 
and  consequent  of  the  other^  the  remaining  terms  will  be 
proportional. 
If  we  have  the  two  proportions,  ♦ 

2  :  6  :  :  8  :  24,    and    2  :  6  :  :  10  :  30, 

we  shall  also  have, 

8  :  24  :  :  10  :  30. 

200.  If  we  have, 

JS        J) 

A  '.  B  '.'.   C  :  B^    and    consequently,    -^  =  -^, 

we  have,  by  dividing  1  by  each  member  of  the  equation, 

A         C 

-=  =       ^    and  consequently,    B  :  A  :  :  B  :   C. 

199.  If  you  have  two  sets  of  proportions  having  an  antecedent  and  coti- 
Dtquent  in  each,  equal  ;  what  will  follow  ? 

200.  If  four  quantities  are  in  proportion,  will  thej  be  In  proportio?' 
when  tftkeu  inveri»elv  ? 


278  ELEMENTARY      ALGEBRA. 

That  is :  Four  proportional  quantities  will  he  i7i  p'opwtion^ 
when  taken  inversely. 

To  give  an  example  in  numbers,  take  the  proportion, 
V  :  14  ::  8  :  16; 
then,  the  inverse  proportion  will  be, 

14  .  7  : :  16  :  8, 
in  which  the  ratio  is  one- half 

201,     The  proportion, 

A  :  B  \\   C  \  I),    gives,    A  y  D  =z  B  x  C, 

To  each  member  of  the  last  equation  add  B  x  D,     We 
shall  then  have, 

{A  ^  B)  X  B  =  (C  +  D)  X  B; 

and  by  separatiiif?  the  factors,  we  obtain,  L      ^      4 

A  ■\-  B  :  B  :i  C  +  B  '.  B,        ^-    '*         ^+^^ 

If,  mstead  of  adding,  we  subtract  B  x  B  from  both 
members,  we  have, 

[A-  B)  X  B  =   [C  -  B)  X  B', 

which  gives, 

A  ~  B  '.  B  '.',   C  -  B  \  B. 

That  is:  If  four  quantities  are  proportional^  they  wiU  be 
in  proportion  by  composition  or  division. 

Thus,  if  we  have  the  proportion, 

9  :  27  :  :  16  :  48, 

201.  If  four  quantities  are  in  proportion,  will  they  be  in  proportion  by 
composition  ?  Will  they  be  in  proportion  by  division  ?  What  ib  tb*> 
^ifTorcncc  between  composition  and  division  ? 


GEOMETRICAL      PEOPORTION.  279 

we  shall  have,  by  composition, 

9  +  27  •  27  :  ;  16  -f  48  :  48 ; 
that  is,  36  :  27  :  :  64  :  48, 

in  which  the  ratio  is  three-fourths. 
The  same  proportion  gives  us,  by  division, 

27  -  9  :  27  ::  48  -  16  :  48; 
that  is,  18  :  27  :  :  32  :  48, 

in  which  the  ratio  is  one  and  one-half. 

202.  If  we  have, 

B  _D 

A-    C 

and  multiply  the  numerator  and  denominator  of  the  first 
member  by  any  number  m,  we  obtain, 

— J   =  -= ,    and    mA  :  mB  :  :  C  :  D, 
mA         C 

Tliat  is :  Equal  multiples  of  two  quantities  have  the  sanie 
ratio  as  the  quantities  themselves. 

For  example,  if  we  have  the  proportion, 

6  :  10  :  :  12  :  24, 

and  multii)ly  the  first  antecedent  and  consequent  by  6,  we 
have, 

30  :  60  :  :  12  :  24, 

in  which  the  ratio  is  still  2. 

203.  The  proportions, 

A  :  B  :  .   C  :  D,    and    A  :  B  : :  B :  JF, 

aoa.  Have  equal  multiples  of  two  quantities  the  aame  ratio  as  the 
quantities  ? 

203.  Suppose  the  antecedent  and  consequent  be  augmented  or  dimiu' 
ifalied  by  quantities  having  the  same  ratio  ? 


280  ELEMENTARY      ALGEBEA. 

give,    A  X  JD  =  B  X  (7,    and    AxF=BxE\ 

adding  and  subtracting  these  equations,  we  obtain, 

A(D±F)  =  B{C^E),    or    A  .  B  .  .  C  ±E  i  D  ±F, 

That  \^'.  If  C  and  Z>,  the  antecedent  and  consequent^  be 
augmented  or  diminished  hy  quantities  E  and  F^  which 
have  the  same  ratio  as  C  to  2)^  the  resulting  quantities  will 
also  have  the  same  ratio. 

Let  us  take,  as  an  example,  the  proportion, 

9  :  18  :  :  20  :  40, 

ill  which  the  ratio  is  2. 

If  we  augment  the  antecedent  and  consequent  by  the 
numbers  15  and  30,  which  have  the  same  ratio,  Ave  shall 
have, 

9  -f-  15  :  18  +  30  :  :  20  :  40; 

that  is,  24  :  48   :  :  20  :  40, 

in  which  the  ratio  is  still  2. 

If  we  diminish  the  second  antecedent  and  consequent  by 
these  numbers  respectively,  we  have, 

9  :  18  •:  20  —  15  :  40  —  30; 

that  is,  9  :  18  :  :  5  :  10, 

in  which  the  ratio  is  till  2. 

204.     If  we  have  several  proportions, 


A  :  B 
A  :  B 
A  :  B 


:  C  :  I),  which  gives  A  x  I)  =  B  x  C, 
:  F  :  F,  which  gives  A  X  F  =  B  x  E^ 
:  G  :  JI,     which  gives    A  x  II  =  B  x  G^ 


&c.,  &c.. 


204.  In  any  number  of  proportions  having  the  same  ratio,  bow  \<'il] 
ony  one  antecedent  be  to  its  consequent  ? 


OEOMKTRICAL      PROPORTION.  281 

we  shall  have,  by  addition, 

A(D  +  F+  IT)  =  B{C  ^-  E'\-  GO; 
and  by  separating  the  factoi-s, 

A  :  B  II  C  ■\-E-\-  G  I  i>+i^4  H, 

That  is:  In  any  7nimher  of  proportio7i8  having  the  same 
rtUio^  any  antecedent  will  be  to  its  consequent  as  the  sum 
of  the  antecedents  to  the  sum  of  the  consequents. 

Let  us  take,  for  example, 

2  :  4  :  :  6  :  12,     and     1  :  2  :  :  3  :  6,    &o. 
Tlien  2:4::64-3:12-f-6; 

that  is,  2  :  4  :  :  9  :  18, 

ill  which  the  ratio  is  still  2. 

a05.     If  we  have  four  proportional  quantities, 

A  :  B  \  :   C  :  J)^    we  have,    -^   =  --^ ; 

and  raising  both  members  to  any  power  whose  exponent  is 
n,  or  extracting  any  root  whose  index  is  w,  we  have, 

-J-  =z  -z^  J     and  consequently. 

That  is:  Jf  four  quajitities  are  proportional,  their  liht 
powers  or  roots  will  be  proportional. 

If  we  have,  for  example, 

2   :  4    :  :  3   :  6, 

we  shall  have  2^  :  4^  :  :  32  :  6^ ; 

806.  In  four  proportional  quantities,  how  aro  like  poweiT  or  roots  ? 


282  ELEMENTARY      ALGEBRA. 

that  is,  4   :  16  :  :  9   :  36, 

ill  which  the  terms  are  proportional,  the  ratio  being  4. 

206,     Let  there  be  two  sets  of  proportions, 

B        J) 

A  :  JB  :  '.   C  :  D,    which  gives      -j  —  -^; 

F       If 

E  \  F  w  G  \  H,    which  gives      -^  =  -^ . 

Multiply  them  together,  member  by  member,  we  have, 

B  X  F  _  JD  X  H 

Ax  E  "   G  X  G' 

A  X  E  \  B  X  F  w  G  X  G  -.  D  x  H. 

That  is :  In  two  sets  of  proportional  quayitities^  iheproduct& 
of  the  corresponding  terms  are  p/roportional. 

Thus,  it*  we  have  the  tuvo  proportions, 


8  :  16 

:  :  10  :     20, 

and, 

3  :     4 

:     6  :       8, 

we  shall  have. 

24  :  64 

:  :  60  :  160. 

GEOMETRICAL     PROGRESSION. 

207.  We  have  thus  far  only  considered  the  case  in  which 
the  ratio  of  the  first  term  to  the  second  is  the  same  as  that 
of  the  third  to  the  fourth. 

206.  In  two  sets  of  proportions,  how  are  the  products  of  the  correspond 
Ing  terras  ? 

207.  What  is  a  geometrical  progression  ?  What  is  the  ratio  of  the 
progression  ?  If  any  term  of  a  progression  be  multiplied  by  the  ratio, 
what  will  the  product  be  ?     If  any  term  be  divided  by  the  ratio,  what 


OBOMKTEIOAL      PBOGBESBION.  2S3 

If  we  have  the  farther  condition,  that  the  ratio  of  the 
second  terra  to  the  third  shall  also  be  the  same  as  that  of 
the  first  to  the  second,  or  of  the  third  to  the  fourth,  we  shall 
have  a  series  of  numbers,  each  one  of  which,  divided  by 
the  preceding  one,  will  give  the  same  ratio.  Hence,  if  any 
term  be  multiplied  by  this  quotient,  the  product  will  be  the 
Bucccoding  term.  A  series  of  numbers  so  formed,  is  called 
a  geometrical  progression.     Hence, 

A  Geometrical  Progression^  or  progression  by  quotients^ 
IS  a  series  of  terms,  each  of  which  is  equal  to  the  preceding 
terra  multiplied  by  a  constant  number^  which  number  is 
called  the  ratio  of  the  progression.    Tims, 

1  :  3  :  9  :  27  :  81  :  243,  &c., 

is  a  geometrical  progression,  in  which  the  ratio  is  3.  It  Is 
written  by  merely  placing  two  dots  between  the  terms. 

Also,  64  :  32  :  16  :  8  :  4  :  2  :  1, 

is  a  geometrical  progression  in  which  the  ratio  is  one-half. 

In  the  first  progression  each  term  is  contained  three  times 
in  the  one  that  follows,  and  hence  the  ratio  is  3.  In  the 
second,  each  term  is  contained  one-half  times  in  the  one 
which  follows,  and  hence  the  rartio  is  one-half. 

The  first  is  called  an  increasing  progression,  and  the 
second  a  decreasing  progression. 

Let  a,  J,  c,  (?,€,/*,  .  .  .  be  numbers,  in  a  progression  by 
quotients  •  they  are  written  thus : 

a:6:c:e?:e:/:<7... 

and  it  is  enunciated  in  the  same  manner  as  a  progression  by 
differen-ces.    It  is  necessary,  however,  to  make  the  distinc* 

will  the  quotient  be  ?  How  \»  a  p'^gression  by  quotients  written  ?  Which 
of  the  terms  is  only  an  antecedent t  Which  only  a  consequent?  How 
nmy  each  of  the  others  be  considered? 


I  obli: 
1  find 


284  ELEMENTARY       ALGEBIJA.. 

tion,  that  one  is  a  series  formed  by  equal  differences,  and 
the  other  a  series  formed  by  equal  quotients  or  ratios.  It 
should  be  remarked  that  each  term  is  at  the  same  time  an 
antecedent  and  a  consequent,  except  the  first,  which  is  only 
an  antecedent,  and  the  last,  which  is  only  a  consequent. 

20§.     Let  r  denote  the  ratio  of  the  progression, 

a  :  b  :  c  :  d  ,  .  . 

7  being  >  1  when  the  progression  is  increasing^  and  r<  1, 
when  it  is  decreasing.    Then,  since, 

h  c  d  e  o~ 

we  have, 

b  =  ar^     c  =  hr  —  ar^,     d  =  cr  =  ar^j     e  =  dr  =  ar*^ 
f  —  er  =  ar^  .  .  . 

that  is,  the  second  term  is  equal  to  «r,  the  third  to  ar^,  the 
fourth  to  ar^,  the  fifth  to  ar*,  &c. ;  and  in  general,  the  nih 
term,  that  is,  one  which  has  ?i  —  1  terms  before  it,  is  ex- 
pressed by  ar''~^. 

Let  I  be  this  term  •  we  then  have  the  formula. 


by  means  of  which  we  can  obtain  any  term  without  being 
ged  to  find  all  the  terms  which  precede  it.     Hence,  to 
the  last  term  of  a  progression,  we  have  the  following 


E  u  L  E . 


I.  liaise  the  ratio  to  a  potcer  whose  exponent  is  one  less 
than  the  numper  of  terms. 

II.  Multiply  the  pov^er  thus  fonnd  by  the  first  term:  the 
lyrodtict  loill  be  the  required  term. 

208.  By  what  letter  do  we  denote  the  ratio  of  a  progression?     In  rd 
Increasing  progression  is  r  greater  or  less  than  1  ?     In  a  df»creasing  pro 


GBOMKTlilCAL      rKOOKEJJBION,  285 

EXAMPLES. 

1 ,  Find  the  6th  term  of  the  progression, 

2  :  4  :  8  :  16  .  .  . 
in  wliich  the  first  term  is  2,  and  the  common  ratio  2. 
6th  term  =  2  x  2*  =  2  x  16  =  32.   Ans. 

2.  Find  the  8th  term  of  the  progression, 

2  :  6  :  18  :  64  .  .  . 

8th  term  =  2  x  3'  =  2  x  2187  =  4374.   Ana. 

8.  Find  the  6th  term  of  the  progression, 
2  :  8  :  32  :  128  .  .  . 
6th  term  =  2  x  4^  =  2  x  1024   =  2048.   An8 

4.  Find  the  7th  term  of  the  progression, 

3  :  9  :  27  :  81  .  .  . 

7th  term  =  3  x  3«  =  3  x  729  =  2187.   Ans, 

5.  Find  the  6th  term  of  the  progression, 

4  :  12  :  sa  :  108  .  .  . 
6th  term  =  4  x  8*  =  4  x  243  =  972.   Ans, 

6.  A  person  agreed  to  pay  his  servant  1  cent  for  the  first 
day,  two  for  the  second,  and  four  for  the  third,  doubling 
every  day  for  ten  days:  how  much  did  he  receive  on  the 
tenth  day?  A71S,   $5.12. 

gresfiion  is  r  greater  or  less  than  1  ?  If  a  is  the  first  term  and  r  the 
ratio,  what  is  the  second  term  equal  to  ?  What  the  third  ?  What  the 
fourth  ?  What  is  the  Uwt  terra  equal  to  ?  Give  the  rule  for  finding  the 
lR8t  term. 


^Wv 


286  ELEMENTARY      ALGEBRA. 

1,  What  is  tlie  8th  term  of  the  progression, 

9  ;  36  :  144  :  576  .  .  . 
8th  term  =   9  X  4'  =   9  X  16384   =   147456.    Aiis, 

8.  Find  the  12th  term  of  the  progression, 
64  :  16  :  4  :  1  :  i  .  .  . 

4 

12th  term  =  64(J)"  =  i!  =  J,  =  ^.  A,^. 


209.    We  will  now  proceed  to  determine  the  smn  of  n 
terms  of  a  progression, 

a  :  b  :  c  :  d  :  e  :  f  :  .  ,  .  :  i  :  k  :  l] 

I  denoting  the  ^ith  term. 

We  have  the  equations  (Art.  208), 

b  =  ar,    c  -—  br^     d  =  cr,    e  =  dr,  .  ,  .  k  =  ir,     I  =  Jcr, 

and  by  adding  them  all  together,  member  to  member,  we 
deduce, 

Sinn  of  l6i  members.  Sum  of  2d  tnenibere. 

b+c+d-{-e-\-  .  .  .  -h7c-\-l={a  +  b  +  c-\-d+  .  .  .  ^.-^^-^>; 

in  which  we  see  that  the  first  member  contains  all  the  terms 
but  «,  and  the  poIjTiomial,  within  the  parenthesis  in  the 
second  member,  contaias  all  the  terras  but  I.  Hence,  if  we 
call  the  sum  of  the  terms  /S,  we  have, 

S^-  a  =  {S  -  l)r  =  Sr  -Ir,     r .  Sr  -  S  =  Ir  -^  a 

whence,  JS  = • 

Kj.A^  r  -  1 

209.  Give  the  rule  for  finding  the  sum  of  the  series.     What  ie  the  first 
step?    What  the  second?    What  the  third? 


GEOMETRICAL      PBOORESSION.  2S7 

Therefore,  to  obtain  the  sum  of  all  the  terras,  or  wim  of  the 
series  of  a  geometrical  progression,  we  have  the 

RULE. 

I.  Multiply  the  last  term  by  the  ratio  : 
n.  Subtract  the  first  term  from  tJie  product : 
in.  Divide  the  remainder  by  the  ratio  diminished  by  1 
and  the  quotient  will  be  the  sum  of  tlie  series, 

1.  Find  the  sum  of  eight  terms  of  the  progression, 

2  :  6  :  18  :  54  :  162  .  .  .  2  X  3'  ;=  4374. 

Ir  -  a        13122  -  2         ^^^^ 

S  = =  r =  6560. 

r  —  \  2 

2.  Find  the  su^i  of  the  progression, 

2  :  4  :  8  :  16  :  32. 

5  =  ?L=^  =  «i^2  ^  62. 
r  —  1  1 

3.  Find  the  sum  of  ten  terms  of  the  progression, 

2  :  6  :  18  :  54  :  162  ...  2  X  33  =  39366. 

Ans.    5904a 

4.  What  debt  may  be  discharged  in  a  year,  or  twelve 
months,  by  pajdng  $1  the  first  month,  $2  the  second  month, 
$4  the  third  month,  and  so  on,  each  succeeding  payment 
being  double  the  last ;  and  what  will  be  the  last  payment ! 


Ans. 


j  Debt,      .      .      14095 


(  Last  payment,  $204^ 

6.  A  daughter  was  married  on  New- Year's  day.  Her 
Cither  gave  her  l5.,  with  an  agreement  to  double  it  on  the 
first  of  the  next  month,  and  at  the  beginning  of  each  succeed- 
ing month  to  double  what  she  had  previously  received.  How 
m'lch  did  she  receive?  Ana,   £204  16/?. 


288  ELEMENTARY      ALGEBRA. 

6.  A  man  bought  ten  bushels  of  wheat,  on  the  condition 
that  he  should  pay  1  cent  for  the  first  bushel,  3  for  the  second, 
9  for  the  third,  aiid  so  on  to  the  last :  what  did  he  pay  for 
the  last  bushel,  and  for  the  ten  bushels  ? 

j  Last  bushel,  |196  83. 
^^'  \  Total  cost,      $295  24. 

7.  A  man  plants  4  bushels  of  barley,  which,  at  the  first 
harvest,  produced  32  bushels ;  these  he  also  plants,  which, 
in  like  manner,  produce  8  fold ;  he  again  plants  all  his  crop, 
and  again  gets  8  fold,  and  so  on  for  16  years:  what  is  his 
last  crop,  and  what  the  sum  of  the  series  ? 

j  Last,  140737488355328  bush. 
^^*  I  Sum,  160842843834660. 

910.  When  the  progression  is  decreasing,  we  have, 
r<  1,  and  Z<  a;  the  above  formula, 

Ir  -  a 

for  the  sum,  is  then  written  under  the  form, 

a  -  Ir 

in  order  that  the  two  terms  of  the  fraction  may  be  positive4 
1.  Find  the  sum  of  the  terms  of  the  progression, 
32  :  16  :  8  :  4  :  2 

32  -  2  X  ^ 

^  =   1^  == ?   =  H  =  62. 

I  —  r  1  1 

2  2 


210.  What  is  the  formula  for  the  sum  of  the  series  of  a  decreasing 
progression  ? 


G  K  u  M  E  T  li  I  C  A  L       P  K  ()  G  R  E  6  8  I  O  N  .  iiyU 

2.  Find  the  sum  of  the  first  twelve  terms  of  the  pro- 

grossion, 

1  /IV'  1 

64  .  10  :  4  :  1  :-:...  :  64(-)   ,     or    - 


65538 


64-—^  X^       256  ^ 


-      a  —  lr               (15536      4                 65536       ,,   .    65635 
S  =  " = =  86  + 


1  —  r  3  3  196008 

4 

ail.  Remark. — We  perceive  that  the  principal  difficulty 
consists  in  obtaining  the  numerical  value  of  the  last  term,  a 
tedious  operation,  even  when  the  number  of  terms  is  not 
very  great. 

3.  Find  the  sum  of  six  terms  of  the  progression, 

612  :  128  :  32  .  .  . 

Ans.    682^ 

4.  Find  the  sura  of  seven  terms  of  the  progression, 

2187  :  729  :  243  .  .  . 

Ans.   327ft 

5.  Fmd  the  sum  of  six  terms  of  the  progression, 

972  :  324  :  108  .  .  . 

Ahs.    1456 

6.  Find  the  sum  of  eight  terms  of  the  progression, 

147456  :  36864  :  9216  .  .  . 

Afis.    190605. 

OP  PROGRESSIONS     HAVTNO   AN    ENPrNlTE    NXTMBEK    OF   TERMS 

213      Let  there  be  the  decreasing  progression, 
a  :  b  :  c  :  d  :  e  :  /  :  ,  .  , 

il2.  When  tho  progression  in  decreasing,  and  the  number  of  terms  in- 
Aoite,  what  is  the  eipreKrion  for  the  value  of  the  sun  of  the  eeries? 
13 


290  ELEMENTARY       ALGEBRA. 

containing  an  indefinite  number  of  terms.     In  the  formula, 

substitute  for  I  its  value,  ar"-\  (Art.  208),  and  we  have, 

„        a  —  ar** 

o  = 


1  -  r   ' 


which  expresses  the  sum  of  n  terms  of  the  progression. 
This  may  be  put  under  the  form, 


„  a  ar* 


W^ 


1  —  r        \  —  r 
Now,  since  the  progression  is  decreasing,  r  is  a  proper 
fraction ;  and  r"  is  also  a  fraction,  which  diminishes  as  n 
increases.     Therefore,  the  greater  the  number  of  terms  we 

take,  the  more  will x  r"  diminish,  and  consequently, 

the  more  will  the  entire  sum  of  all  the  terms  approximate 

to  an  equality  with  the  first  part  of  aS,  that  is,  to • 

Finally,  when  n  is  taken  greater  than  any  given  number, 

or  w  —  mfinity,   then   x  r"   will  be  less  than  any 

given  number,  or  wdll  become  equal  to  0 ;  and  the  expres- 
sion,   ,  will  then  represent  the  true  value  of  the  sum 

of  all  the  terms  of  the  series.  Whence  we  may  conclude, 
lliat  the  expression  for  the  sum  of  the  term,s  of  a  decreasing 
progfession^  in  which  the  number  of  terms  is  iiifinite^  ^5, 

a 

that  is,  equal  to  the  first  term^  divided  by  1  minus  t/ie  ratio. 


GEOMETRICAL      PROGRESSION.  291 

This  is,  properly  speaking,  the  limit  to  which  the  partial 
sums  approach,  as  we  take  a  greater  number  of  terms  in  tho 

progression.    The  difference  between  these  sums  and , 

nay  be  made  as  small  as  we  please,  but  will  only  become 
ixotJiing  when  the  number  of  terms  is  infinite. 

EXAMPLES. 

1.  Find  tlu"  sum  ot 

,       111         !..«.. 
1  •  «  •  ;:  •  ^n,  *  :rr  1    ^o  mnnity. 
3      9      27      81  '  ^ 

We  have,  for  the  expression  of  the  sum  of  the  terms, 

.^  =   -^—  =  -^   =1  =  n.     Ans, 
1  -  r  1         2  ^ 

3 

The  error  committed  by  taking  this  expression  for  the 
value  of  the  sum  of  the  n  first  terms,  is  expressed  by 


1 

^7          

—  r 

2' 

13/  • 

First  take 

n  = 

=  5; 

it  becomes, 

2\3 

J          2.3* 

= 

1 
162 

When  n  = 

=  6, 

we 

find. 

2\3y 

162  ^  3 

= 

1 
486 

3 
Hence,  we  see,  that  the  error  committed  by  taking  _  for 

the  sum  of  a  certain  number  of  terms,  is  less  in  proportion 
EG  this  number  is  greater. 


292  ELEMENTARY        ALGEBRA. 

2.  Again,  take  the  progression, 

1  111  11  o 

We  Lave,    JS  =  - — ~  = =  2.    Am. 

3.  What  is  the  sum  of  the  progression, 

'»  iV  li'  li'  r^o'  &«-'^^i"fi^i^y- 

JS=  -^  =  -i-  =.  il.    ^M.. 

1  —  r  1  9 

10 

213.  In  the  several  questions  of  geometrical  progres 
sion,  there  are  five  numbers  to  be  considered : 

1st.  The  first  t^'rm,  .  .  a. 
2d.  The  ratio,  .  .  .  .  r. 
3d.  The  number  of  terms,  n. 
4th.  Tlie  last  term,  .  .  l. 
5th.  The  sum  of  the  terms,  S. 

214.  We  shall  terminate  this  subject  by  solving  this 
problem : 

To  find  a  mean  proportional  between  any  two  numbers, 
as  m  and  n. 

Denote  the  required  mean  by   x.     We  shah  then  have 
(Art.  197), 

x^  =       rn  X  n  : 


and  hence,  x    =  ^m  x  n. 


218.  How  many  numbers  are  considered  in  a  geometrical  progrefisioii  ? 
What  are  they  ? 
214.  How  do  you  find  a  mean  proportional  between  two  cumbers? 


GEOMETRICAL       PROGKES6ION.  293 

That  is :  Multiply  the  two  numbers  togetJier^  and  extract  the 
square  root  of  the  product. 

1.  What  is  the  geometrical  mean  between  the  numbers 
2  and  8? 

Mean  =   -/S  x  2  =   -/iS   =  4.     Ans, 

2.  What  is  the  mean  between  4  antl  16  ?  Ari8,  ^ 

3.  What  is  the  mean  between  3  and  27  ?  Ans,  9 

4.  Wliut  is  the  mean  between  2  and  72  ?  An9,  12, 

5.  What  is  the  mean  between  4  and  64  ?  Aii^.  16, 


//  y- 


/. 


^y*^    ^ 


294  ELEMENTARY       ALGEBEA. 


CHAl^ER    X. 


OF      LOGARITHMS. 


215.  The  nature  and  properties  of  the  logaritliins  in 
common  use,  will  be  readily  understood  by  considering 
attentively  the  different  powers  of  the  number  10.  They 
are, 


IQO  = 

1 

10^  = 

10 

102  = 

100 

103  = 

1000 

10*  = 

10000 

10*  = 

100000 

&c.. 

&c. 

It  ia  plain  that  the  exponents  0,  1,  2,  3,  4,  5,  <tc.,  form  an 
firithmetical  series  of  which  the  common  difference  is  1 ;  and 
that  the  numbers  1,  10,  100,  1000,  10000,  100000,  &c.,  form 
a  geometrical  progression  of  which  the  common  ratio  is  1 0. 
The  number  10  is  called  the  base  of  the  system  of  logaiithms ; 
and  the  exponents  0,  1,  2,  3,  4,  5,  &c.,  are  the  logarithms  of 

215.  "What  relation  exists  between  the  exponents  1,  2,  8,  &c.  ?  How 
aro  the  corresponding  numbers  10,  100,  1000?  What  is  the  cominoa 
difference  of  the  exponents  ?  What  is  the  common  ratio  of  the  corre- 
sponding numbers  ?  What  is  the  base  of  the  common  system  of  loga« 
rithms  ?  What  are  the  exponents  ?  Of  what  number  is  the  exponent  I 
the  logarithm?    Tie  exponent  2?     The  exponent  3? 


OF      LOOAEITHM8.  *2D5 

the  numbers  which  are  produced  by  raising  10  to  the  powers 
denoted  by  those  exponents. 

216.  If  we  denote  the  logarithm  of  any  number  by  m, 
then  the  number  itself  will  be  the  mth  })ower  of  10  ;  that  is, 
if  we  represent  the  corresponding  number  by  Jif, 

lO""  =  M. 

llius,  if  we  make  m  =  0,  3/ will  be  ec^ual  to  1  ;  if  m  =  1, 
31  will  be  equal  to  10,  &c.     Hence, 

The  logarithm  of  a  number  is  the  exponent  of  t/ie  power 
to  which  it  is  iiecessary  to  raii<e  the  base  of  the  system  in 
order  to  produce  the  number. 

817.  If,  as  before,  10  denotes  the  base  of  the  system 
c»f  logarithms,  m  any  exponent,  and  M  the  corresponduig 
number,  we  shall  then  have, 

lO*"      =       sUy  (1.) 

in  which  m  is  the  logarithm  of  3t. 

If  we  take  a  second  exponent  n,  and  let  N  denote  the 
corresponding  number,  we  shall  have, 

10«  =  iV;  (2.) 

in  which  n  is  the  logarithm  of  N. 

Hi  now,  we  multiply  the  tirst  of  these  etjuations  by  the 
second,  member  by  member,  we  have, 

10-  X   10"  =   lO*"^*  =  J/  X  N\ 

but  since  10  is  the  base  of  the  system,  m  -f  n  is  the  log;v- 
rithm  M  y,  N\  hence, 

216.  If  we  denote  the  base  of  u  system  b>  10,  and  the  exponcut  by 
m,  what  will  represent  the  corresponding  number?  What  is  the  logarithm 
of  a  number  ? 

217.  To  what  is  the  sum  of  the  logarithms  of  any  two  numbers  equal  ? 
Tx)  what,  then,  will  the  addition  cf  logarithms  corre^poud  ? 


296  ELEMENTARY      ALGEBRA. 

The  sum  of  the  logarithms  of  atiy  two  numbers  is  equal 
to  the  logarithm,  of  their  product. 

Therefore,  the  addition  of  logarithms  corresponds  to  the 
multiplication  of  their  numbers. 

218.  If  we  divide  Equation  ( 1 )  by  Equation  (  2  ),  mora- 
ler  by  member,  we  liave, 

10«  M 

but  since  10  is  the  base  of  the  system,  m  —  10  is  the  loga- 
ritlim  of  -^ix.;  hence, 

If  one  number  be  divided  by  another.^  the  logarithm  of 
the  quotient  will  he  equal  to  the  logarithm  of  the  dividend^ 
diminished  by  that  of  the  divisor. 

Therefore,  the  subtraction  of  logarithms  corresponds  to 
the  division  of  their  nuinbers. 

219.  Let  U3  examine  further  the  equations, 


10° 

= 

1 

101 

— 

10 

102 

= 

100 

103 

— 

1000 

&c., 

&c. 

It  is  plain  that  the  logarithm  of  1  is  0,  and  that  the  loga- 
rithra  of  any  number  between  1  and  10,  is  greater  than 


218.  If  one  number  be  divided  by  another,  what  will  the  logarithm 
of  the  quotient  be  equal  to  ?     To  what,  then,  will  the  subtraction  of  loga 

ithms  correspond  ? 

219.  What  is  the  logarithm  of  1?  Between  what  limits  are  the  loga- 
rithms of  ax  numbers  between  1  and  10?  How  are  they  generally  ex- 
pressed  ? 


OF      LOOABITHMS.  297 

0  and  less  than  1.    Tlie  logarithm  is  generally  expressed  by 
decimal  fractions ;  thus, 

log  2   =  0.301030. 

Tlie  logarithm  of  any  nmnber  greater  than  10  and  lese 
than  100,  is  greater  than  1  and  less  than  2,  and  is  expressed 
by  1  and  a  decimal  fraction  ;  thus, 

log  60  =   1.098970. 

The  part  of  the  logarithm  which  stands  at  the  left  of  the 
decimal  point,  is  called  the  characteristic  of  the  logarithm. 
The  characteristic  is  always  07ie  less  than  the  number  of 
places  ofjigicres  in  the  number  whose  logarithm  is  taken. 

Thus,  in  the  first  case,  for  numbers  between  1  and  10, 
there  is  but  one  place  of  figures,  and  the  characteristic  is  0. 
For  numbers  between  10  and  100,  there  are  two  places  of 
figures,  and  the  charactQristic  is  1  ;  and  similarly  for  other 
numbers. 


TABLE   OP   LOGARmnrs. 

220.  A  table  of  logarithms  is  a  table  in  which  are  writ- 
ten the  logarithms  of  all  numbers  between  1  and  some  other 
given  number.  A  table  showing  the  logarithms  of  the 
numbers  between  1  and  100  is  annexed.  The  numbers  are 
written  in  the  column  designated  by  the  letter  N,  and  the 
logarithms  in  the  column  designated  by  Log. 


How  is  it  with  the  logarithms  of  Lumbers  between  10  and  100?  What 
\n  that  part  of  the  logarithm  called  which  stands  at  the  loft  of  the  char 
arterjptic?     What  is  the  value  of  the  characteristic? 

220.  What  is  a  table  of  logarithms  ?  Explain  the  manner  of  finding 
the  logarithns  of  numbers  between  1  and  100? 

13*  — r 

'IF 


298 


ELEMENTARY   ALQEBEA. 


TABLE. 


IT 
1 

Log. 

"N. 

Log. 

IT 

51 

Log. 

r'NT" 

Log. 

0.000000 

26 

1.414973 

1.707570 

76 

1.880814 

2 

0,301030 

27 

1.431364 

52 

1.716003 

77 

1.886491 

3 

0.477121 

28 

1.447158 

53 

1.724276 

78 

1.892095 

i 

0.602060 

29 

1.462398 

54 

1.732394 

79 

1.897627 

5 

0.698970 

30 

1.477121 

55 

1.740363 

80 

1.903090 

6 

0.778151 

31 

1.491362 

56 

1.748188 

81 

1.908485 

7 

0.845098 

32 

1.505150 

57 

1.755875 

82 

1.913814 

8 

0.903090 

33 

1.518514 

58 

1.763428 

83 

1.919078 

9 

0.954243 

34 

1.531479 

69 

1.770852 

84 

1.924279 

10 

1.000000 

35 

1.544068 

60 

1.778151 

85 

1.929419 

11 

1.041393 

36 

1.556303 

61 

1.785330 

86 

1.934498 

12 

1.079181 

37 

1.568202 

62 

1.792392 

87 

1.939519 

13 

1.113943 

38 

1.579784 

63 

1.799341 

88 

1.944483 

14 

1.146128 

39 

1.591065 

64 

1.806180 

89 

1.949390 

15 

1.176091 

40 

1.602060 

65 

1.812913 

90 

1.954243 

16 

1.204120 

41 

1.621784 

66 

1.819544 

91 

1.959041 

17 

1.230449 

42 

1.623249 

67 

1.826075 

92 

1.963788 

18 

1.255273 

43 

1.633468 

68 

1.832509 

93 

1.968483 

19 

1.278754 

44 

1.643453 

69 

1.838849 

94 

1.973128 

20 

1 . 301030 

45 

1.653213 

70 

1.845098 

95 

1.977724 

21 

1.322219 

46 

1.662758 

71 

1.851258 

96 

1.982271 

22 

1.342423 

47 

1.672098 

72 

1.857333 

97 

1.986772 

23 

1.361728 

48 

1.681241 

73 

1.863323 

98 

1.991226 

24 

1.380211 

49 

1.690196 

74 

1.869232 

99 

1.995635 

25 

1.397940 

50 

1.698970 

75 

1.875061 

100 

2.000000 

EXAMPLES. 


1.  Let  it  be  required  to  multiply  8  by  9,  by  means  of 
logarithms.  We  have  seen,  Art.  216,  that  the  sum  of  the 
logarithms  is  equal  to  the  logarithm  of  the  product.  There- 
fore, find  the  logarithm  of  8  from  the  table,  which  is  0.903090, 
and  then  the  logarithm  of  9,  which  is  0.954243  ;  and  their 
sum,  which  is  1.857333,  will  be  the  logarithm  of  the  product. 
In  searching  along  in  the  table,  we  find  that  72  stands  oppo- 
site this  logarithm ;  hence,  72  is  the  product  of  8  by  9, 


OF      LOG  A  KI  TH  M  S. 


2U9 


2.  Wliat  is  the  product  of  7  by  12? 

Logarithm  of    7  is,       . 
Logarithm  of  12  is, 

Logarithm  of  their  product, 
and  the  corresponding  number  is  34. 

3.  What  is  the  product  of  9  b}  11? 

Logarithm  of    9  is,       . 
Logarithm  of  1 1  is, 

Logarithm  of  their  product, 
and  the  corresponding  number  is  99. 


0.845098 
1.079181 

1.924279 


0.954243 
1.041393 

1.99563C 


4.  Let  it  be  required  to  divide  84  by  3.  We  have  seen 
in  Art.  218,  that  the  subtraction  of  Logarithms  corresponds 
to  th<;  division  of  their  numbers.  Hence,  if  we  find  the 
logarithm  of  84,  and  then  subtract  from  it  the  logarithm  of 
8,  the  remainder  will  be  the  logarithm  of  the  quotient. 

The  logarithm  of  84  is,  .         .         .     1.924279 

llie  logarithm  of    3  is,  .         .         .     0.477121 

Their  difference  is,         .         .         .'       .     1.447158 
and  the  corresponding  number  is  28. 

6.  Wliat  is  the  product  of  6  by  7  ? 

Logarithm  of  6  is,         .         .         .         .0.778151 
Logarithm  of  7  is,         .  .     0.845098 

Their  sum  Is,  ....     1.623249 

and  the  corresponding  number  of  the  table,  42. 


The  JVational  Series  of  Standard  School- T^ooK's, 

MATHEMATICS. 


DAVIES'  NATIONAL  COURSE, 


ARITHMETIC. 

1.  Daviea'  Primary  Arithmetic, $  25  $    ^2 

2.  Davies'  Intellectual  Arithmetic, 40  48 

3.  Davies'  Elements  of  Written  Arithmetic, ....  50  60 

4.  Davies'  Practical  Arithmetic, 00  1  00 

Key  to  Practical  Arithmetic, 00 

5.  Davies*  University  Arithmetic, 1  40      1  50 

Key  to  University  Arithmetic, *1  40 

ALGEBRA. 

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GEOMETRY. 

1.  Davies*  Elementary  Geometry  and  Trigonometry,  1  40  1  50 

2.  Davies'  Legendre's  Geometry, 2  25  2  38 

3.  Davies'  Analytical  Geometry  and  Calculus,  ...  2  50  2  63 

4.  Davies'  Descriptive  Geometry, 2  75  2  88 

6.  Davies'  New  Calculus, 2  00 

MENSURATION. 

1.  Davies'  Practical  Mathematics  and  Mensuration, .     1  50      1  60 

2.  Davies'  Elements  of  Surveying, 2  50      2  C:{ 

3.  Davies'  Shades,  Shadows,  and  Perspective,.    .    .    3  75      3  88 

MATHEMATICAL    SCIENCE. 

Davies'  Grammar  of  Arithmetic, *    50 

Davies*  Outlines  of  Mathematical  Science, *1  00 

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17 


The  jVationat  Series  of  Standard  School- ^ooJhs. 

MATHEMATICS-Continued. 

ARITHMETICAL    EXAMPLES. 

Reuck's  Examples  in  Denominate  Numbers  $    no 
Reuck's  Examples  in  Arithmetic i  oo 

These  volumes  differ  from  the  ordinary  arithmetic  in  their  peculiarly 
practical  character.  They  are  composed  mainly  of  examples,  and  afford 
the  moat  severe  and  thorough  discipline  for  the  mind.  While  a  book 
which  should  contain  a  complete  treatise  of  theory  and  practice  would  be 
too  cumbersome  for  every-day  use,  the  iasufficieucy  oi in-actical  examples 
has  been  a  source  of  complaint. 

HIGHER     MATHEMATICS. 

Church's  Elements  of  Calculus 2  50 

Church's  Analytical  Geometry 2  50 

Church's  Descriptive  Geometry,  with  Shades, 

ShadoY/s,  and  PerspectiYe 4  00 

These  volumes  constitute  the  "  West  Point  Course"  in  their  several 
departments. 

Courtenay's  Elements  of  Calculus    .    .    -   -  3  oo 

A  wwrk  especially  popular  at  the  South. 

Hackley's  Trigonometry 2  53 

With  applications  to  navigation  and  surveying,  nautical  and  practical 
geometry  and   geodesy. 

Peck's  Analytical  Geometry l  75 

Peck's  Practical  Calculus i  75 

APPLIED    MATHEMATICS. 

Peck's  Ganot's  Popular  Physics i  75 

Peck's  Elements  of  Mechanics 2  oo 

Peck's  Practical  Calculus i  75 

Peck's  Analytical  Geometry, i  75 

Prof.  W.  G.  Peck,  of  Columbia  College,  has  designed  the  first  of  these  works  for 
the  ordinary  wants  of  schools  in  the  department  of  "Natural  Philosophy.  The 
other  volumes  are  the  briefest  treatises  on  those  subjects  now  published.  Their 
nsethods  are  purely  practical,  and  unembarrassed  by  the  details  which  rather  coa- 
lube  than  simplify  science. 

SLATED    ARITHMETICS. 

This  consists  of  the  application  of  an  artificially  slated  surface  to  the  inner  cover 
of  a  book,  with  flap  of  the  same  opening  outward,  so  that  students  may  refer  to 
the  book  and  use  the  slate  at  one  and  the  same  time,  and  as  though  the  slate  were 
detached.  When  folded  up,  the  slate  preserves  examples  and  memoranda  til' 
needed.  The  material  used  is  as  durable  as  the  stone  slate.  The  additional  cost 
•jf  books  thus  improved  is  trifling. 

20 


One-third  or  one-half  shorter  than  any  other  similar 

course  containing  the  same  amount  of  Knowl' 

edge,  and  thoroughly  Scientific, 


PECK'S 

BRIEF  COURSE  IN  ARITHMETIC, 

By   W.   G.   PECK,    LL.D., 

Professor  of  Mathematics  and  Astronomy  in  Columbia  College,  N.  Y. 

Author  of  "  Analytical  Geometry,"  "  Practical  Calculus," 

*'  Elementary  Mechanics,"  "  Ganot's  Physics." 


The  Theory  of  this  concise  as  well  as  comprehensive 
Course  of  Arithmetic  to  meet  the  wants  of  all  classes  is  as 
follows  : 

I.— FIRST    LESSONS    IN    NUMBERS. 

18mo,  half  bound,        ;  slated, 

This  book  beg^  with  the  simplest  Elementary  Combinations,  illu* 
trating  the  processes  by  suitable  cuts,  but  not  by  pictures  culled  from  the 
primary  readers  and  children's  magazines.  The  true  idea  of  illustration 
is  to  have  a  leading  picture  at  the  head  of  each  part  as  Counting,  Ada . 
tion.  Subtraction,  Multiplication,  Division,  and  Fractions.  The  indivi- 
dual stops  should  be  illustrated  by  diagrams  neatly  engraved  and 
grouped,  and  aiding  in  developing  the  arithmetical  ideas  desired.  This 
style  of  treatment,  covering  50  or  60  pages,  precedes  the  subject  of 
Mental  Arithmetic,  The  book  thus  formed  should  be  all  the  Arithmetic 
needed  to  enter  upon  either  the  Manual  or  the  Complete  Arithmetic.  Its 
place  in  all  schools  would  be  in  dasees  of  pupils  younger  than  about 
twelve  years. 

1 


deck's  jirlt?imetlcat  Course, 


II.— MANUAL   OF   PRACTICAL  ARITHMETIC. 

208  pp.,  18mo,  half  bound,  50/^ ;  slated,  60)^. 

Tliis  book  has  tlie  definitions  clearly  laid  down  (just  as  tliej  are  to 
stand  throughout  the  course) ;  the  rules  too  are  laid  down  exactly  as  they 
are  to  stand  in  all  the  after  course  of  mathematics.  There  is  a  carefully 
illustrated  example  after  each  rule  (illustrated,  that  is,  by  being  wrought 
out  and  explained),  and  then  follows  a  sufficient  number  of  graded  ex- 
amples to  impress  the  rule  on  the  minds  of  the  pupils.  The  place  of 
this  book  would  be  in  the  ordinary  district  schools  where  the  pupils  are 
simply  fitting  themselves  for  the  farm  and  the  workshop,  or  in  graded 
schools  as  a  good  practice  before  entering  on  the  study  of  the  Complete 
Arithmetic.  It  is  adapted  to  children  twelve  to  fourteen  years  of  age, 
and  contains  enough  of  practical  arithmetic  for  common  life.  As  this 
course  of  books  is  chiefly  intended  for  live  teachers,  and  not  so  much 
for  lazy  ones,  such  questions  are  omitted  as,  "If  one  cow  has  two 
horns,  how  many  horns  have  two  cows  ?  "  The  live  teacher,  after  having 
taught  the  First  Lessons,  can  form  enough  of  these  examples  from  the 
objects  around  him,  and  will  do  so. 


III.— THE  COMPLETE  ARITHMETIC. 

318  pp.,  12mo,  half  bound,  90^ ;  slated,  $1.00. 

This  book  contains  everything  necessary  to  a  complete  arithmetician. 
Every  step  is  explained  scientifically.  Every  principle  is  laid  down  in 
clear  language.  Every  rule  is  demonstrated.  A  suitable  number  of 
illustrative  examples  are  given.  In  this  book  pupils  of  intelligence  are 
addressed,  such  as  are  our  children  of  fourteen  years  in  our  average 
schools.   The  book  is  made  consecutive,  logical,  scientific,  concise,  simple. 

A  student  who  follows  this  course  in  the  order  indicated 
will  be  an  Arithmetician  capable  of  making  any  application  of 
his  principles,  and  able  to  give  a  reason  for  the  faith  that  is 
in  him. 

3 


iPecJb^s  Ai'Uhmetical  Course, 


Such  a  course  requires  for  its  full  development  a  live 
teacher — but  in  the  end  the  fruits  will  be  worthy  of  his  labors. 

An  Arithmetical  course  should  be  progressive,  and,  as  fiir  as 
possible,  repetitions  should  be  avoided. 

The  place  for  such  questions  as  a  recent  author  uses  to  usher 
in  his  subjects,  is  in  the  Primary  and  Mental.  To  introduce 
them  into  either  of  the  higher  books  would  be  a  needless  repe- 
tition, and  one  of  our  ablest  teachers  assures  us  that  such  ques- 
tions are  always  passed  over  by  all  good  teachers. 

No  course  in  Arithmetic  can  be  studied  and  mastered  with- 
out much  labor  on  the  part  of  both  pupil  and  teachers,  and 
we  have  yet  to  learn  of  any  plan  by  which  the  subject  can  be 
made  so  easy  that  children  will  cry  for  it. 

With  respect  to  the  outcry  of  keeping  up  to  the  spirit  of  the 
age,  we  will  say  that  the  continually-widening  circle  of  knowl- 
edge demands  that  each  subject  should  be  made  ever  more  and 
more  concise,  more  and  more  abbreviated. 

By  abbreviations  emasculation  is  not  meant,  but  rather 
elimination  of  all  trash  and  superfluous  matter.  The  repeti- 
tion of  primary  principles  in  an  advanced  work,  for  instance, 
and  the  introduction  of  pictures  from  Chatterbox,  are  not  in 
the  direction  of  what  we  may  consider  the  spirit  of  the  age. 

How  well  these  ideas  have  been  carried  out  in  this  course 
wiU  be  determined  by  the  popular  verdict  from  the  great  mass 
of  intelligent  teachers  of  the  country,  and  their  name  is  le- 
gion. We  will  send  specimen  pages  free,  or  copies  for  exam- 
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WITH  SLATES  ATTACHED,  for  $1.00. 

Address 

A.  S.  BARNES  &  CO.,  Publishers, 

NEW    YORK    AND    CHICAGO. 
3 


deck's  A.rithnietical  Course. 


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MATHEMATICS  and  MECHANICS. 

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12mo,  212  pp.,  half  roan.    Price  $1.75. 

DAVIES  &   PECK'S    MATHEMATICAL  DICTIONARY. 

"EVERY    TEACHER    SHOULD     HAVE    A    COPY." 

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branch,  and  of  the  whole  as  forming  a  single  science.  By  Charles  Davies,  LL.D., 
and  Wm.  G.  Peck.    Price  $5.00. 


One  or  more  of  these  works  by  Peck  are  used  in  most  American  Colleges — among 
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LEYAN,  State  University,  Wisconsin,  Capitol  University,  Ohio.  Also  in  the 
Schools  of  Mines  and  Scientific  Schools,  almost  without  exception— such  as  N.  Y. 
School  of  Mines,  Troy  Polytechnic,  Yale  and  Harvard  Scientific  Schools, 
Etc,  Etc. 

clny  or  all  of  the  above  7f07'ks  are  sent,  post-paid,  on  receipt  of  price  ^  or 
at  one-ihird  ojf  to  teachers  for  examination. 

A.   S.   BARNES   &   CO.,  Publishers, 

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The  JVational  Series  of  Standard  Schoot-T^ooks, 

NATURAL    SCIENCE. 

"FOURTEEN  WEEKS"  IN  EACH  BEANCH. 

By   J.   DORMAN   STEELE,  A.  M. 

Steele's  14  Weeks  Course  in  Chemistry  l^,  %\  60 
Steele's  14  Weeks  Course  in  Astronomy  •  i  so 
Steele's  14  Weeks  Course  in  Philosophy  •  i  50 
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Our  Text-Books  In  these  studies  are,  as  a  general  thing,  dull  and  uninteresting. 
They  contain  from  400  to  600  pages  of  dry  facts  and  unconnected  details.  They 
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commences  the  study,  is  confused  by  the  Une  print  and  coarse  print,  and  neither 
knowing  exactly  what  to  Icara  nor  what  to  hasten  over,  is  crowded  through  tliu 
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without  a  clefluitc  and  exact  idea  of  a  single  scientific  principle. 

Steele's  Fourteen  "Weeks  Courses  contain  only  that  which  every  well-informed 
person  should  know,  while  all  that  which  concerns  only  the  professional  scientist 
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is  no  fine  print  to  annoy ;  the  cuts  are  copies  of  genuine  experiments  or  natural 
phenomena,  and  are  of  fine  execution. 

In  fine,  by  a  system  of  condensation  peculiarly  his  own,  the  author  reduces  each 
branch  to  the  limits  of  a  single  term  of  study,  while  sacrificing  nothing  that  is  es- 
sential, and  nothing  that  is  usually  retained  from  the  study  of  the  larger  manuals 
in  common  use.  Thus  the  student  has  rare  opportunity  to  economiu  hit  time,  or 
rather  to  employ  that  which  he  has  to  the  best  advantage, 

A  notable  feature  la  the  author's  charming  "  style,"  fortified  by  an  enthusiasm 
over  his  subject  in  which  the  student  will  not  fail  to  partake.  Believing  tlrnt 
Natural  Science  is  full  of  Ihsclnatlon,  he  has  moulded  it  into  a  form  that  attract* 
vho  attention  and  kindles  the  enthusiasm  of  the  pupil. 

The  recent  editions  contain  the  author's  *'  Practical  Questions"  on  a  plan  never 
before  attempted  in  scientific  text-books.  These  are  questions  as  to  the  nature 
and  cause  of  common  phenomena,  and  are  not  directly  answered  in  the  text,  the 
design  being  to  test  and  promote  an  intelligent  use  of  the  student's  knowledge  of 
(the  foregoing  principles. 

Steele's  General  Key  to  his  Works-    •    •    .  *i  53 

This  work  is  mainly  composed  of  Answers  to  the  Practical  Questions  and  Solu- 
tions of  the  Problems  in  the  author's  celebrated  "  Fourteen  Weeks  Courses  "  in 
the  several  sciences,  with  many  hinti  to  teachers,  minor  Tables,  &c.  Should  bn 
on  every  teacher's  desk. 

34 


Thi  JVational  Series  of  Standard  School-Hooks. 

MODERN  LANGUAGE, 

French  and  English  Primer, $    lo 

German  and  English  Primer, lo 

Spanish  and  English  Primer, :i3 

The  names  of  common  objects  properl7  illustrated  and  arranged  in  sasy 

lessons. 

Ledru's  French  Fables, 75 

Ledru's  French  Grammar, i  oo 

Ledru's  French  Reader, .    .    .   p l  oo 

The  author's  long  experience  has  enabled  him  to  present  the  most  thor- 
oughly practical  text-books  extant,  in  this  branch.  The  system  of  pro- 
nunciation (by  phonetic  illustration)  is  origiaal  with  this  author,  and  wiil 
commend  itself  to  all  American  teachers,  as  it  enables  their  pupils  to  se- 
cure an  absolutely  correct  pronunciation -without  the  assistance  of  a  nativa 
master.  This  feature  is  peculiarly  valuable  also  to  "  self-taught"  students. 
The  directions  for  ascertaining  the  gender  of  French  nouns — also  a  great 
etumbling-block — are  peculiar  to  this  work,  and  will  be  found  remarkably 
competent  to  the  end  proposed.  The  criticism  of  teachers  and  the  test  of 
tke  school-room  is  invited  to  this  excellent  series,  with,  confidenco. 

Worman's  French  Echo, i  25 

To  teach  conversational  French  by  actual  practice,  on  an  entirely  new- 
plan,  which  recognizes  the  importance  of  the  student  learning  to  think  in 
the  language  which  he  speaks.  It  furnishes  an  extensive  vocabulary  of 
words  and  expressions  in  common  use,  and  suffices  to  free  the  learner 
from  the  embarrassments  which  the  peculiarities  of  his  own  tongue  are 
likely  to  be  to  him,  and  to  make  him  thoroughly  familiar  with  the  use 
of  proper  idioms. 

Worman's  German  Echo,  . 1  2'i 

On  the  same  plan.    See  Worman's  German  Series,  page  42. 

Pujol's  Complete  French  Class-Book,  ...  2  23 

Offers,  in  one  volume,  methodically  arrayed,  a  complete  French  course 
—usually  embraced  in  serius  of  from  five  to  twelve  books,  including  tha 
bulky  and  expensive  Lexicon.  Here  are  Grammar,  Conversation,  and 
choice  Literature — selected  from  the  best  French  authors.  Each  branch 
is  Uioroughly  handled  ;  and  the  student,  having  diligently  completed  tha 
course  as  prescribed,  may  consider  himself,  without  further  application, 
aufait  in  the  most  polite  and  elegant  languago  of  modern  times. 

Maurice-Poitevin's  Grammaire  Francaise,  •  i  00 

American  schools  are  at  last  supplied  with  an  American  edition  of  this 
famous  text-book.  Many  of  our  best  institutions  have  for  years  been  pro- 
curing it  from  abroad  rather  than  forego  the  advantages  it  offers.  Tho 
policy  of  putting  students  who  have  acquired  some  proficiency  from  tho 
ordinary  text-books,  into  a  Grammar  written  in  the  vernacular,  can  not 
■fee  too  highly  commended.  It  afi"ords  an  opportunity  for  finish  and  review 
at  once ;  while  embodying  abundant  practice  of  its  own  rules. 

Joynes'  French  Pronunciation, 30 

Willard's  Historia  de  los  Estados  Unidos, .  2  00 

The  History  of  the  United  States,  translated  by  Professors  Tolon  and 
Db  Tobnos,  will  be  found  a  valuable,  instructive^  and  enter taiiiinjj  read- 
ing-book fv>r  Spanish  classes.  ^  ^ 


The  ^aiionat  Teachers'  Zibrary. 


TEACHERS'  LIBRARY. 


Object  Lessons-Welch **i  oc 

This  is  a  complete  exposition  of  the  poi 
"  object  teaching/'  for  teachers  of  primary  cl 


This  iB  a  complete  exposition  of  the  popular  modem  eystem  of 
"    '        "*      "     *  ;iat 


Theory  and  Practice  of  Teaching— Page  •    .  *i  so 

This  volume  has,  without  doubt,  been  read  by  two  himdred  thousand 
teachers,  and  its  popularity  remains  undiralnii<hed— large  editions 
being  exhausted  yearly.  It  was  the  pioneer,  as  it  is  now  the  patri- 
arch of  professional  works  for  teachers. 

The  Graded  School-Wells *i  25 

The  proper  way  to  organize  traded  schools  is  here  illustrated.  The 
author  has  availed  hicieelf  of  tue  best  elements  of  the  several  systems 
prevalent  in  Boston,  New  York,  Philadelphia,  Cincinnati,  St.  Louis, 
and  other  cities. 

The  Normal— Holbrook n  50 

Carries  a  working  school  on  its  visit  to  teachers,  showing  the  most 
apptoved  methods  of  teaching  all  the  common  branches,  Includluer  the 
tecnnicalitics,  explanations,  demonstrations,  and  definitions  Intro- 
ductory and  peculiar  to  each  branch. 

The  Teachers'  Institute— Fowie *i  25 

This  is  a  volume  of  suggestions  inspired  by  the  author's  experience 
at  Institutes,  In  the  Instruction  of  jeung  teachers.  A  thousand  poinu 
of  interest  to  tliis  class  are  most  satisfactorily  dealt  with. 

Schools  and  Schoolmasters— Dickens  •   •   .  *i  25 

Appropriate  selections  from  the  writings  of  the  great  novelist 

The  Metric  System— Davies *i  50 

Considered  with  reference  to  its  general  Introtlnction,  and  embrac- 
ing the  views  of  John  Quincy  Adams  and  Sir  John  Uerschel. 

The  Student ;— The  Educator— Phelps    •  cach,*i  50 
The  Discipline  of  Life-Phelps *i  75 

The  authoress  of  these  works  is  one  of  the  most  distinguished 
writers  on  education  ;  and  they  cannot  fail  to  prove  a  valunl)Iti  addi- 
tion to  the  School  and  Teachers'  Libraries,  being  In  a  high  degree 
both  Interesting  and  Instructive. 

A  Scientific  Basis  of  Education— Hecker  •    .  *2  50 

Adaptation  of  study  and  classification  by  temperaments. 

4d 


The  JVational  Teachers'  JLibrary. 


Liberal  Education  of  Women— Orton    .    •    *Si  so 

Treate  of  "  the  demand  and  the  method ;"  being  a  compilation  of  the  best  and 
most  advanced  thought  on  this  subject,  by  the  leading  writers  and  educators  in 
England  and  America.    Edited  by  a  Professor  in  Vassar  College. 

Education  Abroad— Northrop *i  so 

A  thorough  discussion  of  the  advantages  and  disadvantages  of  sending  American 
children  to  Europe  to  be  educated ;  also,  Papers  on  Legal  Prevention  of  Illiteracy, 
Study  and  Health,  Labor  as  an  Educator,  and  other  kindred  subjects.  By  the  Hon. 
Secretary  of  Education  for  Connecticut. 

The  Teacher  and  the  Parent— Northend  •    .  ''i  so 

A  treatise  upon  common-school  education,  designed  to  lead  teachers  to  view  their 
ealling  in  its  true  light,  and  to  stimulate  them  to  fidelity. 

The  Teachers'  Assistant— Northend  ....  *i  50 

A  natural  continuation  of  the  author's  previous  work,  more  directly  calculated  for 
daily  use  in  the  administration  of  school  discipline  and  instruction. 

School  Government— Jewell *i  so 

Full  of  advanced  ideas  on  the  subject  which  its  title  indicates.  The  criticisms 
upon  current  theories  of  punishment  and  schemes  of  administration  have  excited 
general  attention  and  comment. 

Grammatical  Diagrams— Jewell *i  oo 

The  diagram  system  of  teaching  grammar  explained,  defended,  and  improved. 
The  curious  in  literature,  the  searcher  for  truth,  those  interested  in  new  inventions, 
as  well  as  the  disciples  of  Prof.  Clark,  who  would  see  their  favorite  theory  fairly 
treated,  all  want  this  book.  There  are  many  who  would  like  to  be  made  familiar 
with  this  system  before  risking  its  use  in  a  class.    The  opportunity  is  here  afibrded. 

The  Complete  Examiner— Stone *i  ^s 

Consists  of  a  series  of  questions  on  every  English  branch  of  school  and  academic 
instruction,  with  reference  to  a  given  page  or  article  of  leading  text-books  where 
the  answer  may  be  found  in  full.  Prepared  to  aid  teachers  in  securing  certificates, 
pupils  in  preparing  for  promotion,  and  teachers  in  selecting  review  questions. 

School  Amusements— Root *i  so 

To  assist  teachers  in  making  the  school  interesting,  with  hints  upon  the  manage- 
ment of  the  school-room.  Rules  for  military  and  gymnastic  exercises  are  included. 
Illustrated  by  diagrams. 

Institute  Lectures— Bates *i  so 

These  lectures,  originally  delivered  before  institutes,  are  based  upon  various 
topics  in  the  dejjartments  of  mental  and  moral  culture.  The  volume  is  calculated 
to  prepare  the  will,  awaken  the  inquiry,  and  stimulate  the  thought  of  the  zealous 
teacher. 

Method  of  Teachers'  Institutes— Bates    •    •    •    *7S 

Sets  forth  the  best  method  of  conducting  institutes,  with  a  detailed  account  of  the 
object,  organization,  plan  of  instruction,  and  true  theory  of  education  on  which 
such  instruction  should  be  based. 

History  and  Progress  of  Education  ....  *i  so 

The  systems  of  education  prevailing  in  all  nations  and  ages,  the  gradual  advance 
to  the  present  time,  and  the  bearing  of  the  past  upon  the  present  in  this  regard,  art 
worthy  of  the  careful  investigation  of  all  concerned  in  education. 

49 


yjitf  J^aMonai  Teachers*  Zibrary, 


American  Education— Mansfield $i  50 

A  treatise  on  the  principles  and  elements  of  educ 
this  cotiutry,  with  ideas  towards  distinctive  republic 


A  treatise  on  the  principles  and  elements  of  educatioTi,  as  practiced  in 
^  with  ideas  towards  distinctive  republican  and  Christian  edu- 


American  Institutions— De  Tocqueville   .    .*i  so 

A  Talaablo  Index  to  the  genius  of  our  Government. 

Universal  Education— Mayhew *i  75 

The  Biibjoct  is  approached  with  the  clear,  keen  perception  of  one  who 
has  obsorved  its  iiec^ssity,  and  realized  its  feasibility  and  expediency 
%like.  The  redoeiiiinK  and  elevating  power  of  iniprured  coiiinion  schools 
constitutes  the  inspiration  of  the  volume. 

Higher  Christian  Education— Dwighl  .    .    .*i  60 

A  treatise  on  the  principles  and  spirit,  the  modes,  directions,  and  ra- 
snlts  of  all  true  teaching;  showing  that  right  education  should  appeal  to 
every  element  of  enthusiasm  in  the  teacher's  naturo. 

Oral  Training  Lessons— Barnard  .   .   .   .    *i  oo 

The  object  of  this  very  iiseftil  work  is  to  funiieh  materiel  for  instrnc- 
tors  to  impart  orally  to  their  classes,  in  branchee  not  ii;  lU'lly  taught  in 
coinmoo  schools,  embmcing  all  departments  of  Natural  &;ience  and 
much  general  knowled<;e. 

Lectures  on  Natural  History— Chadbourne    *  75 

Affording  many  themes  for  oral  mstmction  in  this  inrerestinjj  science — 
in  schools  where  it  is  not  pursued  an  a  clasd  exercise. 


Outlines  of  Mathematical  Science— Davies  *i  oo 

A  manual  snggesting  the  best  methods  of  presenting  mathematical  in- 
structiuti  on  the  part  of  the  teacher,  with  that  comprehensive  view  of  the 
whole  which  is  necessary  to  the  intelligent  treatment  of  a  part,  in  science. 

Nature  &  Utility  of  Mathematics— Davies  •     *!  50 

An  elaborate  and  lucid  exposition  of  the  principles  which  lie  at  the 
Foundation  nf  pure  mathematics,  with  a  hi;?hly  ingenious  application  of 
their  results  to  the  development  of  the  easential  iJc-a  of  t)ie  different 
branches  of  the  science. 

Mathematical  Dictionary- Davies  &  Peck  .*5  oo 

Tl>is  cyclopaedia  of  mathematical  science  defines  with  completeness, 
precision,  and  accuracy,  every  technical  terra,  tlius  constituting  a  popular 
traatiss  on  each  branch,  and  a  general  view  of  the  whole  subject 

School  Architecture— Barnard *2  25 

Attention  is  here  called  to  the  vital  connection  between  a  pood  schooW 
ho  ise  and  a  good  8cbot>l,  with  plans  and  speciiiOiUons  UiX  securing  lb* 
(ormtir  in  the  moMt  eeoooioical  and  satisfactory  oianner. 

60 


JVatlonal  School  library. 


THE  SCHOOL  LIBRARY. 

The  two  elements  of  instruction  and  entertainment  were  never  more  happily  com- 
bined than  in  this  collection  of  standard  books.  Children  and  adults  alike  will  here 
find  ample  food  for  the  mind,  of  the  sort  that  is  easily  digested,  while  not  degener- 
ating to  the  level  of  modern  romance. 


LIBRARY   OF  LITERATURE. 
Milton's  Paradise  Lost.     Boyd's  illustrated  Ed.,  $1  60 

Young's  Night  Thoughts  .    .    .    .  do.  .  .  i  eo 

Cowper's  Task,  Table  Talk,  &c.    .  do.  .  .  i  eo 

Thomson's  Seasons do.  .  .  i  eo 

Pollok's  Course  of  Time   .    .    .    .  do.  .  .  i  eo 

These  works,  models  of  the  best  and  purest  literature,  are  beautifully  illustrated, 
and  notes  explain  all  doubtful  meanings. 

Lord  Bacon's  Essays  (Boyd's  Edition)    .    .    .    i  eo 

Another  grand  English  classic,  affording  the  highest  example  of  purity  in  lan- 
guage and  ir'cyle. 

The  Iliad  of  Homer.     Translated  by  Pope.    .     .         80 

Those  who  are  unable  to  read  this  greatest  of  ancient  writers  in  the  original, 
should  not  fail  to  avail  themselves  of  this  metrical  version. 

Compendium  of  Eng.  Literature— Cleveland,  2  50 
English  Literature  of  XlXth  Century  do.  2  50 
Compendium  of  American  Literature    do.        2  50 

Nearly  one  hundred  and  fifty  thousand  volumes  of  Prof  Cleveland's  inimitable 
compendiums  have  been  sold.  Taken  together  they  present  a  complete  view  of 
literature.  To  the  man  who  can  afford  but  a  few  books  these  will  supply  the  place 
of  an  extensive  library.  From  commendations  of  the  very  highest  authorities  the 
following  extracts  will  give  some  idea  of  the  enthusiasm  with  which  the  works  are 
regarded  by  scholars : 

With  the  Bible  and  your  volumes  one  mi^ht  leave  libraries  without  very  painful 
regret. — The  work  cannot  be  found  from  whict^n  the  same  limits  so  much  interest- 
ing and  valuable  information  may  be  obtained.  —  Good  taste,  fine  scholarship, 
familiar  acquaintance  with  literature,  unwearied  industry,  tact  acquired  by  practice, 
an  interest  in  the  culture  of  the  young,  and  regard  for  truth,  purity,  philanthropy 
and  religion  are  united  in  Mr.  Cleveland. — A  judgment  clear  and  impartial,  a  taste 
at  once  delicate  and  severe. — The  biographies  are  just  and  discriminating. — An 
admirable  bird's-eye  view.— Acquaints  the  reader  with  the  characteristic  method, 
tone,  and  quality  of  each  writer. —Succinct,  carefully  written,  and  wonderfully  com- 
prehensive in  detail,  etc.,  etc. 

Milton's  Poetical  Works— Cleveland   ...    2  50 

This  is  the  very  best  edition  of  the  great  Poet.  It  includes  a  life  of  the  author, 
notes,  dissertations  on  each  poem,  a  faultless  text,  and  is  the  only  edition  of  Milton 
with  a  complete  verbal  Index. 

51 


"F9UR1EEN  WEEKS"  lyUTURIlL  SCIENCE 

RIEB^    TRE^^nSE    IN    EA.CII    J3RA, 

J.  QOHMAH  STEELE,  A.M. 


14 


.  NATURAL  PHILOSOPHY, 
WMLU  1  ASTRONOMY. 

COURSES 


CHEMISTRY, 

GEOLOGY. 

The-e  volumes  constitute  the  most  RTallable,  practical,  and  attractive  text-books  on 
the  Scieuoes  ever  pabUehed.  Each  volume  may  be  completed  in  a  single  term  of  stodj. 

THE  FAMOUS  PRACTICAL   QUESTIONS 
devleed  by  this  anthor  are  alone  eufflclent  to  place  his  books  In  every  Academy  and 
Grammar  School  of  the  land.    These  are  questions  as  to  the  nature  and  cause  or  com- 
mon phenomena,  and  are  not  directly  answered  in  the  text,  the  design  being  to  test 
and  promote  an  intelligent  use  of  the  student's  knowledge  of  the  foregoing  prmciples. 

TO  MAKE  SCIENCE  POPULAR 
Ib  a  prime  object  of  these  books.    To  this  end  each  subject  is  invested  with  a  charm- 
inf  interest  oy  the  peculiarly  happy  use  of  language  and  illustration  in  which  thia 
author  excels. 

THEIR  HEA  VY  PREDECESSORS 
demand  as  much  of  the  student's  time  for  the  acquisition  of  the  principles  of  a  single 
branch  as  these  for  the  whole  course. 

PUBLIC  APPRECIA  TION. 
The  author's  great  success  in  meeting  an  urgent,  popular  need.  Is  Indicated  by  the 
fiujt  (probably  unparalleled  in  the  history  of  scientific  text-books),  that  although  the 
first  volume  was  issued  in  1867,  the  yearly  sale  is  already  at  the  rate  of 

PHYSIOLOGY   AND   HEALTH^ 

By  EDWARD  JARVIS,  M.D. 
ELFJIEMS  OF  PHYSIOLOGY, 
PHYSIOLOGY  AIVD  LAWS  OF  HEALTH. 

The  only  books  extant  which  approach  this  subject  ■with  a  proper  view  of  the  true 
object  of  teaching  Physiology  In  schools,  viz.,  that  scholars  may  know  how  to  take 
care  of  their  own  health.    The  child  instructed  from  these  works  will  be  always 

^'^onaidc^  the  lilies." 

BOTANY. 

WOOD'S  AMERICAN  BOTANIST  AND  FLORIST. 

This  new  and  eagerly  expected  work  Is  the  result  of  the  author's  experience  and 
life-long  labors  in 

CLASSIFYING   THE  SCIENCE  OF  BOTANY. 
He  has  at  lensrth  attained  the  realization  of  his  hopes  br  a  wonderfully  ingenious  pro- 
cess of  condensation  and  arrangement,  and  presents  to  the  world  in  this  single  modern 
ate-siised  volume  a  COMPL.ETE  MANUAL. 
In  870  duodecimo  pages  be  has  actually  recorded  and  defined 

NEA  RL  Y  4,ooo  SPECIES. 
The  treatises  on  Descriptive  and  Structural  Botanv  are  models  of  concise  statement, 
whicn  leave  nothing  to  be  said.    Of  entirely  new  features,  the  most  notable  are  the 
Synoptical  Tables  for  the  blackboard^  and  the  distinction  of  species  and  rarleties  by 
yariation  in  the  type. 

FroC  Wood,  by  this  work,  establishes  •  Just  claim  to  his  title  of  the  great 

AMERICAN  EXPONENT  OF  BOTANY. 


JlRwIb 


fS®    WBSf    l^@liri    ©@WSi 

And  Only  Thorough  and  Complete  Mathematical  Series 


lisr     TKCIiEB     I>uA.R,TS- 


/.   COMMON  SCHOOL   COURSE. 

Davies'  Primary  Arithmetic- -—The  fundamental,  principles  displa 

Object  Lessons. 
Davies'  Intellectual  Arithmetic— Heferrtng  all  operations  to  the  ui 

the  only  tangible  basis  for  logical  development. 
Davies'  Slements  of  Written  Arithmeticr— A  practical  introdnc 

the  whole  subject.    Theory  subordinated  to  Practice. 
Davies'  Practical  Arithmetic-*— The  most  successful  combination  of ' 

and  Practice,  clear,  exact,  brief,  and  comprehensive. 

//.  ACADEMIC  COURSE. 

Davies'  University  Arithmetic-*— Treating  the  subject  exhausth 

a  science,  in  a  logical  series  of  connected  propositions. 
Davies'  Elementary  Algebra-*— A  connecting  link,  conducting  the 

easily  from  arithmetical  processes  to  abstract  analysis. 
Davies'  University  Alg-ebra-* — For  institutions  desiring  a  more  co 

but  not  the  fullest  course  in  pure  Algebra. 
Davies'  Practical  Mathematics-— The  science  practically  applied 
•-v^^  useful  arts,  as  Drawing,  Architecture,  Surveying,  Mechanics,  etc. 
Davies'  Elementary  Geometry-— The  important  principles  in  simple 

but  with  all  the  exactness  of  vigorous  reasoning* 
Davies'  Elements  of  Surveying--— Re-writtea  in  1870.     The  simple 

most  practical  presentation  for  youths  of  12  to  IG. 

///.  COLLEGIATE  COURSE. 

Davies'  Bourdon's  Al g-ebra -*— Embracing  Sturm's   Theorem,  and  a 

exliauBtive  and  scholarly  course. 
Davies'  University  Alg-ebra-*- A  shorter  course  than  Bourdon,  for  I 

tions  have  less  time  to  give  the  subject. 
Davies'  XiOgendrc'^s  Geometry-— Acknowledged  A^onfy  satisfactory  t 

of  its  grade.    300,000  copies  have  been  sold. 
Davies'  Analytical  Geometry  and  Calculus-— The  shorter  tre 

combined  in  one  volume,  are  more  available  for  American  courses  of  study. 
Davies'  Analytical  Geometry-  \  The  original  compendiums,  for  the 
Davies'  Diff-  &  Int-  Calculus-  '  siring  to  give  full  time  to  each  brai 
Davies'  Descriptive  Geometry.— With  application  to  Spherical  Trigo 

try,  Spherical  Projections,  and  Warped  Surfaces. 
Davies'  Shades,  Shado-vrs,  and  Perspective-— A  succinct  exposit 

the  mathematiqal  principles  involved. 
Davies'  Science  of  BHathematics-— For  teachers,  embracing 

I.  Grammar  of  Abithmbtic,  III.  Logic  and  Utilitt  op  Mathb 

IL  OUTLINE3  OF  MATHEMATICS,  IV.  MaTHEMATIC  Ali  DICTIONARY. 


KEYS    MAY    BE     OBTAINED    PROM     THE    PtTBT-lSHEBS 

BY   TEACHERS    CKLY. 


^U  ^fu,  all  pnuatris,  and  all  Jiinrs. 

fAL       ITTQ'T'n'DV      STANDARD 
I        IlljJ  1 UH  1 1    TEXT-BOOKS. 


is  (Philosophy  teaching  by  Examples^ 


IINITFn    STIITES        ^    Souths   History  of   the 

UI1I  I  UU     W  I  «  I  UW.  UNITED  STATES.    By  JAanes 

rmthor  of  the  National  Geographical  Series.    An  elementary  work 
itecbetical  plan,  with  Maps,  Engravings,  Memoriter  Tables,  etc    For 
ycjungeet  pupils. 

ard's  School  History,  for  Grammar  Schools  and  Academic  classei. 
\si\cCi  to  cultivate  the  memory,  the  intellect,  and  the  taste,  and  to  sow  tho 
1-  of  yi-^""  >>v  rrinteiuDlatiou  of  the  actions  of  the  good  and  great. 


?>'(. 


im 


in 


I 


926515 

THE  UNIVERSITY  OF  CALIFORNIA  UBRARY 


-tii!^' and  rhi\;ilrou>  hi!«tory,  profiiscly  Illustrated.  \\i;li  th--  ]i  .'.i.iis 
i  (loti>)tful  i)()rti()ii<  t-o  iiitr("lui«'d  as  not  to  deceive,  whil>' adding  cxtcudcd 
•.v.n  to  thcBubject. 

FRfil      Willard's  Universal  History,   a  vast  subject eo  arranged 

^  »•"*-■  and  illnstrated  as  to  be  Icse  difflcalt  to  acquire  or  retain.  Its 

'    --.  !n  fiwt,  is  Bommarized  on  one  pa^*^,  In  a  grand  "Temple  of 

f  Nations. 

ral  Summary  of  History.  Belnfr  the  Summaries  of  American,  and 
r  iirlish  and  French  Jllstory,  boond  in  one  volume.  The  leading  events  lu 
aistorles  of  theee  three  nationa  epitomized  is  the  briefest  manner. 


S.   BARNES   &   CO., 


